List of trigonometric identities#Triple-angle formulae

{{Main|Pythagorean trigonometric identity}}

File:Trigonometric functions and their reciprocals on the unit circle.svg

The basic relationship between the sine and cosine is given by the Pythagorean identity:

$$\backslash sin^2\backslash theta\; +\; \backslash cos^2\backslash theta\; =\; 1,$$

where $\backslash sin^2\; \backslash theta$ means $\{(\backslash sin\; \backslash theta)\}^2$ and $\backslash cos^2\; \backslash theta$ means $\{(\backslash cos\; \backslash theta)\}^2.$

This can be viewed as a version of the Pythagorean theorem, and follows from the equation $x^2\; +\; y^2\; =\; 1$ for the unit circle. This equation can be solved for either the sine or the cosine:

$$\backslash begin\{align\}$$

\sin\theta &= \pm \sqrt{1 - \cos^2\theta}, \\

\cos\theta &= \pm \sqrt{1 - \sin^2\theta}.

\end{align}

where the sign depends on the quadrant of $\backslash theta.$

Dividing this identity by $\backslash sin^2\; \backslash theta$, $\backslash cos^2\; \backslash theta$, or both yields the following identities:

$$\backslash begin\{align\}$$

&1 + \cot^2\theta = \csc^2\theta \\

&1 + \tan^2\theta = \sec^2\theta \\

&\sec^2\theta + \csc^2\theta = \sec^2\theta\csc^2\theta

\end{align}

Using these identities, it is possible to express any trigonometric function in terms of any other (up to a plus or minus sign):

By examining the unit circle, one can establish the following properties of the trigonometric functions.

File:Unit Circle - symmetry.svg

When the direction of a Euclidean vector is represented by an angle $\backslash theta,$ this is the angle determined by the free vector (starting at the origin) and the positive $x$-unit vector. The same concept may also be applied to lines in an Euclidean space, where the angle is that determined by a parallel to the given line through the origin and the positive $x$-axis. If a line (vector) with direction $\backslash theta$ is reflected about a line with direction $\backslash alpha,$ then the direction angle $\backslash theta^\{\backslash prime\}$ of this reflected line (vector) has the value

$$\backslash theta^\{\backslash prime\}\; =\; 2\; \backslash alpha\; -\; \backslash theta.$$

The values of the trigonometric functions of these angles $\backslash theta,\backslash ;\backslash theta^\{\backslash prime\}$ for specific angles $\backslash alpha$ satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function. These are also known as {{em|reduction formulae}}.{{harvnb|Selby|1970|loc=p. 188}}

File:Unit Circle - shifts.svg

The sign of trigonometric functions depends on quadrant of the angle. If and {{math|sgn}} is the sign function,

$$\backslash begin\{align\}$$

\sgn(\sin \theta) = \sgn(\csc \theta) &= \begin{cases}

+1 & \text{if}\ \ 0 < \theta < \pi \\

-1 & \text{if}\ \ {-\pi} < \theta < 0 \\

0 & \text{if}\ \ \theta \in \{0, \pi \}

\end{cases}

\\[5mu]

\sgn(\cos \theta) = \sgn(\sec \theta) &= \begin{cases}

+1 & \text{if}\ \ {-\tfrac12\pi} < \theta < \tfrac12\pi \\

-1 & \text{if}\ \ {-\pi} < \theta < -\tfrac12\pi \ \ \text{or}\ \ \tfrac12\pi < \theta < \pi\\

0 & \text{if}\ \ \theta \in \bigl\{{-\tfrac12\pi}, \tfrac12\pi \bigr\}

\end{cases}

\\[5mu]

\sgn(\tan \theta) = \sgn(\cot \theta) &= \begin{cases}

+1 & \text{if}\ \ {-\pi} < \theta < -\tfrac12\pi \ \ \text{or}\ \ 0 < \theta < \tfrac12\pi \\

-1 & \text{if}\ \ {-\tfrac12\pi} < \theta < 0 \ \ \text{or}\ \ \tfrac12\pi < \theta < \pi \\

0 & \text{if}\ \ \theta \in \bigl\{{-\tfrac12\pi}, 0, \tfrac12\pi, \pi \bigr\}

\end{cases}

\end{align}

The trigonometric functions are periodic with common period $2\backslash pi,$ so for values of {{mvar|θ}} outside the interval $(\{-\backslash pi\},\; \backslash pi],$ they take repeating values (see {{slink|#Shifts and periodicity}} above).

{{See also|Proofs of trigonometric identities#Angle sum identities|Small-angle approximation#Angle sum and difference}}

File:Angle sum.svg

File:Diagram showing the angle difference trigonometry identities for sin(a-b) and cos(a-b).svg

These are also known as the {{em|angle addition and subtraction theorems}} (or {{em|formulae}}).

$$\backslash begin\{align\}$$

\sin(\alpha + \beta) &= \sin \alpha \cos \beta + \cos \alpha \sin \beta \\

\sin(\alpha - \beta) &= \sin \alpha \cos \beta - \cos \alpha \sin \beta \\

\cos(\alpha + \beta) &= \cos \alpha \cos \beta - \sin \alpha \sin \beta \\

\cos(\alpha - \beta) &= \cos \alpha \cos \beta + \sin \alpha \sin \beta

\end{align}

The angle difference identities for $\backslash sin(\backslash alpha\; -\; \backslash beta)$ and $\backslash cos(\backslash alpha\; -\; \backslash beta)$ can be derived from the angle sum versions by substituting $-\backslash beta$ for $\backslash beta$ and using the facts that $\backslash sin(-\backslash beta)\; =\; -\backslash sin(\backslash beta)$ and $\backslash cos(-\backslash beta)\; =\; \backslash cos(\backslash beta)$. They can also be derived by using a slightly modified version of the figure for the angle sum identities, both of which are shown here.

These identities are summarized in the first two rows of the following table, which also includes sum and difference identities for the other trigonometric functions.

When the series $\backslash sum\_\{i=1\}^\backslash infty\; \backslash theta\_i$ converges absolutely then

$$\backslash begin\{align\}$$

{\sin}\biggl(\sum_{i=1}^\infty \theta_i\biggl)

&= \sum_{\text{odd}\ k \ge 1} (-1)^\frac{k-1}{2} \!\!

\sum_{\begin{smallmatrix} A \subseteq \{\,1,2,3,\dots\,\} \\

\left|A\right| = k\end{smallmatrix}}

\biggl(\prod_{i \in A} \sin\theta_i \prod_{i \not \in A} \cos\theta_i\biggr) \\

{\cos}\biggl(\sum_{i=1}^\infty \theta_i\biggr)

&= \sum_{\text{even}\ k \ge 0} (-1)^\frac{k}{2} \,

\sum_{\begin{smallmatrix} A \subseteq \{\,1,2,3,\dots\,\} \\ \left|A\right| = k\end{smallmatrix}}

\biggl(\prod_{i \in A} \sin\theta_i \prod_{i \not \in A} \cos\theta_i\biggr) .

\end{align}

Because the series $\backslash sum\_\{i=1\}^\backslash infty\; \backslash theta\_i$ converges absolutely, it is necessarily the case that $\backslash lim\_\{i\; \backslash to\; \backslash infty\}\; \backslash theta\_i\; =\; 0,$ $\backslash lim\_\{i\; \backslash to\; \backslash infty\}\; \backslash sin\; \backslash theta\_i\; =\; 0,$ and $\backslash lim\_\{i\; \backslash to\; \backslash infty\}\; \backslash cos\; \backslash theta\_i\; =\; 1.$ In particular, in these two identities an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are cofinitely many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero.

When only finitely many of the angles $\backslash theta\_i$ are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.

Let $e\_k$ (for $k\; =\; 0,\; 1,\; 2,\; 3,\; \backslash ldots$) be the {{mvar|k}}th-degree elementary symmetric polynomial in the variables

$$x\_i\; =\; \backslash tan\; \backslash theta\_i$$

for $i\; =\; 0,\; 1,\; 2,\; 3,\; \backslash ldots,$ that is,

$$\backslash begin\{align\}$$

e_0 &= 1 \\[6pt]

e_1 &= \sum_i x_i &&= \sum_i \tan\theta_i \\[6pt]

e_2 &= \sum_{i

e_3 &= \sum_{i

&\ \ \vdots &&\ \ \vdots

\end{align}

Then

$$\backslash begin\{align\}$$

{\tan}\Bigl(\sum_i \theta_i\Bigr)

&= \frac{{\sin}\bigl(\sum_i \theta_i\bigr) / \prod_i \cos \theta_i}

{{\cos}\bigl(\sum_i \theta_i\bigr) / \prod_i \cos \theta_i} \\[10pt]

&= \frac

{\displaystyle

\sum_{\text{odd}\ k \ge 1} (-1)^\frac{k-1}{2}

\sum_{

\begin{smallmatrix} A \subseteq \{1,2,3,\dots\} \\

\left|A\right| = k\end{smallmatrix}}

\prod_{i \in A} \tan\theta_i}

{\displaystyle

\sum_{\text{even}\ k \ge 0} ~ (-1)^\frac{k}{2} ~~

\sum_{

\begin{smallmatrix} A \subseteq \{1,2,3,\dots\} \\

\left|A\right| = k\end{smallmatrix}}

\prod_{i \in A} \tan\theta_i}

= \frac{e_1 - e_3 + e_5 -\cdots}{e_0 - e_2 + e_4 - \cdots} \\[10pt]

{\cot}\Bigl(\sum_i \theta_i\Bigr)

&= \frac{e_0 - e_2 + e_4 - \cdots}{e_1 - e_3 + e_5 -\cdots}

\end{align}

using the sine and cosine sum formulae above.

The number of terms on the right side depends on the number of terms on the left side.

For example:

$$\backslash begin\{align\}$$

\tan(\theta_1 + \theta_2) &

= \frac{ e_1 }{ e_0 - e_2 }

= \frac{ x_1 + x_2 }{ 1 \ - \ x_1 x_2 }

= \frac{ \tan\theta_1 + \tan\theta_2 }{ 1 \ - \ \tan\theta_1 \tan\theta_2 },

\\[8pt]

\tan(\theta_1 + \theta_2 + \theta_3) &

= \frac{ e_1 - e_3 }{ e_0 - e_2 }

= \frac{ (x_1 + x_2 + x_3) \ - \ (x_1 x_2 x_3) }{ 1 \ - \ (x_1x_2 + x_1 x_3 + x_2 x_3) },

\\[8pt]

\tan(\theta_1 + \theta_2 + \theta_3 + \theta_4) &

= \frac{ e_1 - e_3 }{ e_0 - e_2 + e_4 } \\[8pt] &

= \frac{ (x_1 + x_2 + x_3 + x_4) \ - \ (x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4) }{ 1 \ - \ (x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4) \ + \ (x_1 x_2 x_3 x_4) },

\end{align}

and so on. The case of only finitely many terms can be proved by mathematical induction.{{cite conference |last=Bronstein |first=Manuel |title=Simplification of real elementary functions |pages=207–211 |doi=10.1145/74540.74566 |book-title=Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation |editor-first= G. H. |editor-last=Gonnet |conference=ISSAC '89 (Portland US-OR, 1989-07) |location=New York |publisher=ACM |year=1989 |isbn=0-89791-325-6}} The case of infinitely many terms can be proved by using some elementary inequalities.Michael Hardy. (2016). "On Tangents and Secants of Infinite Sums." The American Mathematical Monthly, volume 123, number 7, 701–703. https://doi.org/10.4169/amer.math.monthly.123.7.701

$$\backslash begin\{align\}$$

{\sec}\Bigl(\sum_i \theta_i \Bigr) &= \frac{\prod_i \sec\theta_i}{e_0 - e_2 + e_4 - \cdots} \\[8pt]

{\csc}\Bigl(\sum_i \theta_i \Bigr) &= \frac{\prod_i \sec\theta_i }{e_1 - e_3 + e_5 - \cdots}

\end{align}

where $e\_k$ is the {{mvar|k}}th-degree elementary symmetric polynomial in the {{mvar|n}} variables $x\_i\; =\; \backslash tan\; \backslash theta\_i,$ $i\; =\; 1,\; \backslash ldots,\; n,$ and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left.{{cite journal |last=Hardy |first=Michael |year=2016 |title=On Tangents and Secants of Infinite Sums |journal=American Mathematical Monthly |volume=123 |issue=7 |pages=701–703 |doi=10.4169/amer.math.monthly.123.7.701 |url=https://zenodo.org/record/1000408 }} The case of only finitely many terms can be proved by mathematical induction on the number of such terms.

For example,

$$\backslash begin\{align\}$$

\sec(\alpha+\beta+\gamma)

&= \frac{\sec\alpha \sec\beta \sec\gamma}

{1 - \tan\alpha\tan\beta - \tan\alpha\tan\gamma - \tan\beta\tan\gamma} \\[8pt]

\csc(\alpha+\beta+\gamma)

&= \frac{\sec\alpha \sec\beta \sec\gamma}

{\tan\alpha + \tan\beta + \tan\gamma - \tan\alpha\tan\beta\tan\gamma}.

\end{align}

{{Main|Ptolemy's theorem}}

{{See also|History of trigonometry#Classical antiquity}}

File:Diagram illustrating the relation between Ptolemy's theorem and the angle sum trig identity for sin.svg

Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved. It states that in a cyclic quadrilateral $ABCD$, as shown in the accompanying figure, the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference trigonometric identities.{{cite web | url=https://www.cut-the-knot.org/proofs/sine_cosine.shtml | title=Sine, Cosine, and Ptolemy's Theorem }} The relationship follows most easily when the circle is constructed to have a diameter of length one, as shown here.

By Thales's theorem, $\backslash angle\; DAB$ and $\backslash angle\; DCB$ are both right angles. The right-angled triangles $DAB$ and $DCB$ both share the hypotenuse $\backslash overline\{BD\}$ of length 1. Thus, the side $\backslash overline\{AB\}\; =\; \backslash sin\; \backslash alpha$, $\backslash overline\{AD\}\; =\; \backslash cos\; \backslash alpha$, $\backslash overline\{BC\}\; =\; \backslash sin\; \backslash beta$ and $\backslash overline\{CD\}\; =\; \backslash cos\; \backslash beta$.

By the inscribed angle theorem, the central angle subtended by the chord $\backslash overline\{AC\}$ at the circle's center is twice the angle $\backslash angle\; ADC$, i.e. $2(\backslash alpha\; +\; \backslash beta)$. Therefore, the symmetrical pair of red triangles each has the angle $\backslash alpha\; +\; \backslash beta$ at the center. Each of these triangles has a hypotenuse of length $\backslash frac\{1\}\{2\}$, so the length of $\backslash overline\{AC\}$ is $2\; \backslash times\; \backslash frac\{1\}\{2\}\; \backslash sin(\backslash alpha\; +\; \backslash beta)$, i.e. simply $\backslash sin(\backslash alpha\; +\; \backslash beta)$. The quadrilateral's other diagonal is the diameter of length 1, so the product of the diagonals' lengths is also $\backslash sin(\backslash alpha\; +\; \backslash beta)$.

When these values are substituted into the statement of Ptolemy's theorem that $|\backslash overline\{AC\}|\backslash cdot\; |\backslash overline\{BD\}|=|\backslash overline\{AB\}|\backslash cdot\; |\backslash overline\{CD\}|+|\backslash overline\{AD\}|\backslash cdot\; |\backslash overline\{BC\}|$, this yields the angle sum trigonometric identity for sine: $\backslash sin(\backslash alpha\; +\; \backslash beta)\; =\; \backslash sin\; \backslash alpha\; \backslash cos\; \backslash beta\; +\; \backslash cos\; \backslash alpha\; \backslash sin\; \backslash beta$. The angle difference formula for $\backslash sin(\backslash alpha\; -\; \backslash beta)$ can be similarly derived by letting the side $\backslash overline\{CD\}$ serve as a diameter instead of $\backslash overline\{BD\}$.

File:Visual demonstration of the double-angle trigonometric identity for sine.svg

Formulae for twice an angle.{{harvnb|Selby|1970|loc=pg. 190}}

{{startplainlist}}

• $\backslash sin\; (2\backslash theta)\; =\; 2\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash theta\; =\; (\backslash sin\; \backslash theta\; +\backslash cos\; \backslash theta)^2\; -\; 1\; =\; \backslash frac\{2\; \backslash tan\; \backslash theta\}\; \{1\; +\; \backslash tan^2\; \backslash theta\}$
• $\backslash cos\; (2\backslash theta)\; =\; \backslash cos^2\; \backslash theta\; -\; \backslash sin^2\; \backslash theta\; =\; 2\; \backslash cos^2\; \backslash theta\; -\; 1\; =\; 1\; -\; 2\; \backslash sin^2\; \backslash theta\; =\; \backslash frac\{1\; -\; \backslash tan^2\; \backslash theta\}\; \{1\; +\; \backslash tan^2\; \backslash theta\}$
• $\backslash tan\; (2\backslash theta)\; =\; \backslash frac\{2\; \backslash tan\; \backslash theta\}\; \{1\; -\; \backslash tan^2\; \backslash theta\}$
• $\backslash cot\; (2\backslash theta)\; =\; \backslash frac\{\backslash cot^2\; \backslash theta\; -\; 1\}\{2\; \backslash cot\; \backslash theta\}\; =\; \backslash frac\{1\; -\; \backslash tan^2\; \backslash theta\}\; \{2\; \backslash tan\; \backslash theta\}$
• $\backslash sec\; (2\backslash theta)\; =\; \backslash frac\{\backslash sec^2\; \backslash theta\}\{2\; -\; \backslash sec^2\; \backslash theta\}\; =\; \backslash frac\{1\; +\; \backslash tan^2\; \backslash theta\}\; \{1\; -\; \backslash tan^2\; \backslash theta\}$
• $\backslash csc\; (2\backslash theta)\; =\; \backslash frac\{\backslash sec\; \backslash theta\; \backslash csc\; \backslash theta\}\{2\}\; =\; \backslash frac\{1\; +\; \backslash tan^2\; \backslash theta\}\; \{2\; \backslash tan\; \backslash theta\}$

{{endplainlist}}

Formulae for triple angles.

{{startplainlist}}

• $\backslash sin\; (3\backslash theta)\; =3\backslash sin\backslash theta\; -\; 4\backslash sin^3\backslash theta\; =\; 4\backslash sin\backslash theta\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{3\}\; -\backslash theta\backslash right)\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{3\}\; +\; \backslash theta\backslash right)$
• $\backslash cos\; (3\backslash theta)\; =\; 4\; \backslash cos^3\backslash theta\; -\; 3\; \backslash cos\backslash theta\; =4\backslash cos\backslash theta\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{3\}\; -\backslash theta\backslash right)\backslash cos\backslash left(\backslash frac\{\backslash pi\}\{3\}\; +\; \backslash theta\backslash right)$
• $\backslash tan\; (3\backslash theta)\; =\; \backslash frac\{3\; \backslash tan\backslash theta\; -\; \backslash tan^3\backslash theta\}\{1\; -\; 3\; \backslash tan^2\backslash theta\}\; =\; \backslash tan\; \backslash theta\backslash tan\backslash left(\backslash frac\{\backslash pi\}\{3\}\; -\; \backslash theta\backslash right)\backslash tan\backslash left(\backslash frac\{\backslash pi\}\{3\}\; +\; \backslash theta\backslash right)$
• $\backslash cot\; (3\backslash theta)\; =\; \backslash frac\{3\; \backslash cot\backslash theta\; -\; \backslash cot^3\backslash theta\}\{1\; -\; 3\; \backslash cot^2\backslash theta\}$
• $\backslash sec\; (3\backslash theta)\; =\; \backslash frac\{\backslash sec^3\backslash theta\}\{4-3\backslash sec^2\backslash theta\}$
• $\backslash csc\; (3\backslash theta)\; =\; \backslash frac\{\backslash csc^3\backslash theta\}\{3\backslash csc^2\backslash theta-4\}$

{{endplainlist}}

Formulae for multiple angles.{{Cite web |last=Weisstein |first=Eric W. |title=Multiple-Angle Formulas |url=https://mathworld.wolfram.com/Multiple-AngleFormulas.html |access-date=2022-02-06 |website=mathworld.wolfram.com |language=en}}

{{startplainlist}}

• $\backslash begin\{align\}$

\sin(n\theta) &= \sum_{k\text{ odd}} (-1)^\frac{k-1}{2} {n \choose k}\cos^{n-k} \theta \sin^k \theta =

\sin\theta\sum_{i=0}^{(n+1)/2}\sum_{j=0}^{i} (-1)^{i-j} {n \choose 2i + 1}{i \choose j}

\cos^{n-2(i-j)-1} \theta \\

{}&=\sin(\theta)\cdot\sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}(-1)^k\cdot {(2\cdot \cos(\theta))}^{n-2k-1}\cdot {n-k-1 \choose k} \\

{}&=2^{(n-1)} \prod_{k=0}^{n-1} \sin(k\pi/n+\theta)

\end{align}

\sum_{i=0}^{n/2}\sum_{j=0}^{i} (-1)^{i-j} {n \choose 2i}{i \choose j} \cos^{n-2(i-j)} \theta \\

{} &= \sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor} (-1)^k\cdot {(2\cdot \cos(\theta))}^{n-2k}\cdot {n-k \choose k}\cdot\frac{n}{2n-2k}

\end{align}

• $\backslash cos((2n+1)\backslash theta)=(-1)^n\; 2^\{2n\}\backslash prod\_\{k=0\}^\{2n\}\backslash cos(k\backslash pi/(2n+1)-\backslash theta)$
• $\backslash cos(2\; n\; \backslash theta)=(-1)^n\; 2^\{2n-1\}\; \backslash prod\_\{k=0\}^\{2n-1\}\; \backslash cos((1+2k)\backslash pi/(4n)-\backslash theta)$
• $\backslash tan(n\backslash theta)\; =\; \backslash frac\{\backslash sum\_\{k\backslash text\{\; odd\}\}\; (-1)^\backslash frac\{k-1\}\{2\}\; \{n\; \backslash choose\; k\}\backslash tan^k\; \backslash theta\}\{\backslash sum\_\{k\backslash text\{\; even\}\}\; (-1)^\backslash frac\{k\}\{2\}\; \{n\; \backslash choose\; k\}\backslash tan^k\; \backslash theta\}$

{{endplainlist}}

The Chebyshev method is a recursive algorithm for finding the {{mvar|n}}th multiple angle formula knowing the $(n-1)$th and $(n-2)$th values.{{cite web|last=Ward|first=Ken|website=Ken Ward's Mathematics Pages|title=Multiple angles recursive formula|url=http://trans4mind.com/personal_development/mathematics/trigonometry/multipleAnglesRecursiveFormula.htm}}

$\backslash cos(nx)$ can be computed from $\backslash cos((n-1)x)$, $\backslash cos((n-2)x)$, and $\backslash cos(x)$ with

$$\backslash cos(nx)=2\; \backslash cos\; x\; \backslash cos((n-1)x)\; -\; \backslash cos((n-2)x).$$

This can be proved by adding together the formulae

$$\backslash begin\{align\}$$

\cos ((n-1)x + x) &= \cos ((n-1)x) \cos x-\sin ((n-1)x) \sin x \\

\cos ((n-1)x - x) &= \cos ((n-1)x) \cos x+\sin ((n-1)x) \sin x

\end{align}

It follows by induction that $\backslash cos(nx)$ is a polynomial of $\backslash cos\; x,$ the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials#Trigonometric definition.

Similarly, $\backslash sin(nx)$ can be computed from $\backslash sin((n-1)x),$ $\backslash sin((n-2)x),$ and $\backslash cos\; x$ with

$$\backslash sin(nx)=2\; \backslash cos\; x\; \backslash sin((n-1)x)-\backslash sin((n-2)x)$$

This can be proved by adding formulae for $\backslash sin((n-1)x+x)$ and $\backslash sin((n-1)x-x).$

Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:

$$\backslash tan\; (nx)\; =\; \backslash frac\{\backslash tan\; ((n-1)x)\; +\; \backslash tan\; x\}\{1-\; \backslash tan\; ((n-1)x)\; \backslash tan\; x\}\backslash ,.$$

$$\backslash begin\{align\}$$

\sin \frac{\theta}{2} &= \sgn\left(\sin\frac\theta2\right) \sqrt{\frac{1 - \cos \theta}{2}} \\[3pt]

\cos \frac{\theta}{2} &= \sgn\left(\cos\frac\theta2\right) \sqrt{\frac{1 + \cos\theta}{2}} \\[3pt]

\tan \frac{\theta}{2}

&= \frac{1 - \cos \theta}{\sin \theta}

= \frac{\sin \theta}{1 + \cos \theta}

= \csc \theta - \cot \theta

= \frac{\tan\theta}{1 + \sec{\theta}} \\[6mu]

&= \sgn(\sin \theta) \sqrt\frac{1 - \cos \theta}{1 + \cos \theta}

= \frac{-1 + \sgn(\cos \theta) \sqrt{1+\tan^2\theta}}{\tan\theta} \\[3pt]

\cot \frac{\theta}{2}

&= \frac{1 + \cos \theta}{\sin \theta}

= \frac{\sin \theta}{1 - \cos \theta}

= \csc \theta + \cot \theta

= \sgn(\sin \theta) \sqrt\frac{1 + \cos \theta}{1 - \cos \theta} \\

\sec \frac{\theta}{2}

&= \sgn\left(\cos\frac\theta2\right) \sqrt{\frac{2}{1 + \cos\theta}} \\

\csc \frac{\theta}{2}

&= \sgn\left(\sin\frac\theta2\right) \sqrt{\frac{2}{1 - \cos\theta}} \\

\end{align}

{{AS ref|4, eqn 4.3.20-22|72}}{{MathWorld|title=Half-Angle Formulas|urlname=Half-AngleFormulas}}

Also

$$\backslash begin\{align\}$$

\tan\frac{\eta\pm\theta}{2} &= \frac{\sin\eta \pm \sin\theta}{\cos\eta + \cos\theta} \\[3pt]

\tan\left(\frac{\theta}{2} + \frac{\pi}{4}\right) &= \sec\theta + \tan\theta \\[3pt]

\sqrt{\frac{1 - \sin\theta}{1 + \sin\theta}} &= \frac{\left|1 - \tan\frac{\theta}{2}\right|}{\left|1 + \tan\frac{\theta}{2}\right|}

\end{align}

{{See also|Tangent half-angle formula}}

These can be shown by using either the sum and difference identities or the multiple-angle formulae.

The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a compass and straightedge construction of angle trisection to the algebraic problem of solving a cubic equation, which allows one to prove that trisection is in general impossible using the given tools.

A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation {{math|1=4x³ − 3x + d = 0}}, where $x$ is the value of the cosine function at the one-third angle and {{mvar|d}} is the known value of the cosine function at the full angle. However, the discriminant of this equation is positive, so this equation has three real roots (of which only one is the solution for the cosine of the one-third angle). None of these solutions are reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots.

Obtained by solving the second and third versions of the cosine double-angle formula.

{{stack |float=left |File:Diagram showing how to derive the power reduction formula for cosine.svg }}

{{stack |float=left |File:Diagram showing how to derive the power reducing formula for sine.svg }}

{{clear}}

In general terms of powers of $\backslash sin\; \backslash theta$ or $\backslash cos\; \backslash theta$ the following is true, and can be deduced using De Moivre's formula, Euler's formula and the binomial theorem.

==Product-to-sum and sum-to-product identities==

File:visual_proof_prosthaphaeresis_cosine_formula.svg ]]

The product-to-sum identitiesAbramowitz and Stegun, p. 72, 4.3.31–33 or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems. Historically, the first four of these were known as Werner's formulas, after Johannes Werner who used them for astronomical calculations.{{Cite book |last=Eves |first=Howard |title=An introduction to the history of mathematics |date=1990 |publisher=Saunders College Pub |isbn=0-03-029558-0 |edition=6th |location=Philadelphia |page=309 |oclc=20842510}} See amplitude modulation for an application of the product-to-sum formulae, and beat (acoustics) and phase detector for applications of the sum-to-product formulae.

{{startplainlist}}

• $\backslash begin\{align\}$

\cos \theta\, \cos \varphi &= \tfrac12\bigl(\!\!~\cos(\theta - \varphi) + \cos(\theta + \varphi)\bigr) \\[3mu]

\sin \theta\, \sin \varphi &= \tfrac12\bigl(\!\!~\cos(\theta - \varphi) - \cos(\theta + \varphi)\bigr) \\[3mu]

\sin \theta\, \cos \varphi &= \tfrac12\bigl(\!\!~\sin(\theta + \varphi) + \sin(\theta - \varphi)\bigr) \\[3mu]

\cos \theta\, \sin \varphi &= \tfrac12\bigl(\!\!~\sin(\theta + \varphi) - \sin(\theta - \varphi)\bigr)

\end{align}

• $\backslash tan\; \backslash theta\backslash ,\; \backslash tan\; \backslash varphi\; =\backslash frac\{\backslash cos(\backslash theta-\backslash varphi)-\backslash cos(\backslash theta+\backslash varphi)\}\{\backslash cos(\backslash theta-\backslash varphi)+\backslash cos(\backslash theta+\backslash varphi)\}$
• $\backslash tan\; \backslash theta\backslash ,\; \backslash cot\; \backslash varphi\; =\; \backslash frac\{\backslash sin(\backslash theta\; +\; \backslash varphi)\; +\; \backslash sin(\backslash theta\; -\; \backslash varphi)\}\{\backslash sin(\backslash theta\; +\; \backslash varphi)\; -\; \backslash sin(\backslash theta\; -\; \backslash varphi)\}$

& \text{where }e = (e_1,\ldots,e_n) \in S=\{1,-1\}^n

\end{align}

\prod_{k=1}^n \sin\theta_k=\frac{(-1)^{\left\lfloor\frac

{n}{2}\right\rfloor}}{2^n}\begin{cases}

\displaystyle\sum_{e\in S}\cos(e_1\theta_1+\cdots+e_n\theta_n)\prod_{j=1}^n e_j \;\text{if}\; n\; \text{is even},\\

\displaystyle\sum_{e\in S}\sin(e_1\theta_1+\cdots+e_n\theta_n)\prod_{j=1}^n e_j \;\text{if}\; n\; \text{is odd}

\end{cases}

{{endplainlist}}

File:Diagram illustrating sum to product identities for sine and cosine.svg

The sum-to-product identities are as follows:Abramowitz and Stegun, p. 72, 4.3.34–39

{{startplainlist}}

• $\backslash sin\; \backslash theta\; \backslash pm\; \backslash sin\; \backslash varphi\; =\; 2\; \backslash sin\backslash left(\; \backslash frac\{\backslash theta\; \backslash pm\; \backslash varphi\}\{2\}\; \backslash right)\; \backslash cos\backslash left(\; \backslash frac\{\backslash theta\; \backslash mp\; \backslash varphi\}\{2\}\; \backslash right)$
• $\backslash cos\; \backslash theta\; +\; \backslash cos\; \backslash varphi\; =\; 2\; \backslash cos\backslash left(\; \backslash frac\{\backslash theta\; +\; \backslash varphi\}\; \{2\}\; \backslash right)\; \backslash cos\backslash left(\; \backslash frac\{\backslash theta\; -\; \backslash varphi\}\{2\}\; \backslash right)$
• $\backslash cos\; \backslash theta\; -\; \backslash cos\; \backslash varphi\; =\; -2\backslash sin\backslash left(\; \backslash frac\{\backslash theta\; +\; \backslash varphi\}\{2\}\backslash right)\; \backslash sin\backslash left(\backslash frac\{\backslash theta\; -\; \backslash varphi\}\{2\}\backslash right)$
• $\backslash tan\backslash theta\backslash pm\backslash tan\backslash varphi=\backslash frac\{\backslash sin(\backslash theta\backslash pm\; \backslash varphi)\}\{\backslash cos\backslash theta\backslash ,\backslash cos\backslash varphi\}$

{{endplainlist}}

{{Main|Hermite's cotangent identity}}

Charles Hermite demonstrated the following identity.{{cite journal|first=Warren P. |last=Johnson |title=Trigonometric Identities à la Hermite |journal=American Mathematical Monthly |volume=117 |issue=4 |date=Apr 2010 |pages=311–327 |doi=10.4169/000298910x480784|s2cid=29690311 }} Suppose $a\_1,\; \backslash ldots,\; a\_n$ are complex numbers, no two of which differ by an integer multiple of {{pi}}. Let

$$A\_\{n,k\}\; =\; \backslash prod\_\{\backslash begin\{smallmatrix\}\; 1\; \backslash le\; j\; \backslash le\; n\; \backslash \backslash \; j\; \backslash neq\; k\; \backslash end\{smallmatrix\}\}\; \backslash cot(a\_k\; -\; a\_j)$$

(in particular, $A\_\{1,1\},$ being an empty product, is 1). Then

$$\backslash cot(z\; -\; a\_1)\backslash cdots\backslash cot(z\; -\; a\_n)\; =\; \backslash cos\backslash frac\{n\backslash pi\}\{2\}\; +\; \backslash sum\_\{k=1\}^n\; A\_\{n,k\}\; \backslash cot(z\; -\; a\_k).$$

The simplest non-trivial example is the case {{math|1=n = 2}}:

$$\backslash cot(z\; -\; a\_1)\backslash cot(z\; -\; a\_2)\; =\; -1\; +\; \backslash cot(a\_1\; -\; a\_2)\backslash cot(z\; -\; a\_1)\; +\; \backslash cot(a\_2\; -\; a\_1)\backslash cot(z\; -\; a\_2).$$

For coprime integers {{mvar|n}}, {{mvar|m}}

$$\backslash prod\_\{k=1\}^n\; \backslash left(2a\; +\; 2\backslash cos\backslash left(\backslash frac\{2\; \backslash pi\; k\; m\}\{n\}\; +\; x\backslash right)\backslash right)\; =\; 2\backslash left(\; T\_n(a)+\{(-1)\}^\{n+m\}\backslash cos(n\; x)\; \backslash right)$$

where {{mvar|T_n}} is the Chebyshev polynomial.{{citation needed|date=October 2023}}

The following relationship holds for the sine function

$$\backslash prod\_\{k=1\}^\{n-1\}\; \backslash sin\backslash left(\backslash frac\{k\backslash pi\}\{n\}\backslash right)\; =\; \backslash frac\{n\}\{2^\{n-1\}\}.$$

More generally for an integer {{math|n > 0}}{{cite web |title=Product Identity Multiple Angle |url=https://math.stackexchange.com/q/2095330 }}

$$\backslash sin(nx)\; =\; 2^\{n-1\}\backslash prod\_\{k=0\}^\{n-1\}\; \backslash sin\backslash left(\backslash frac\{k\}\{n\}\backslash pi\; +\; x\backslash right)\; =\; 2^\{n-1\}\backslash prod\_\{k=1\}^\{n\}\; \backslash sin\backslash left(\backslash frac\{k\}\{n\}\backslash pi\; -\; x\backslash right).$$

or written in terms of the chord function $\backslash operatorname\{crd\}x\; \backslash equiv\; 2\backslash sin\backslash tfrac12x$,

$$\backslash operatorname\{crd\}(nx)\; =\; \backslash prod\_\{k=1\}^\{n\}\; \backslash operatorname\{crd\}\backslash left(\backslash frac\{k\}\{n\}2\backslash pi\; -\; x\backslash right).$$

This comes from the factorization of the polynomial $z^n\; -\; 1$ into linear factors (cf. root of unity): For any complex {{mvar|z}} and an integer {{math|n > 0}},

$$z^n\; -\; 1\; =\; \backslash prod\_\{k=1\}^\{n\}\backslash left(\; z\; -\; \backslash exp\backslash Bigl(\backslash frac\{k\}\{n\}2\backslash pi\; i\backslash Bigr)\backslash right).$$

For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different phase shifts is also a sine wave with the same period or frequency, but a different phase shift. This is useful in sinusoid data fitting, because the measured or observed data are linearly related to the {{mvar|a}} and {{mvar|b}} unknowns of the in-phase and quadrature components basis below, resulting in a simpler Jacobian, compared to that of $c$ and $\backslash varphi$.

The linear combination, or harmonic addition, of sine and cosine waves is equivalent to a single sine wave with a phase shift and scaled amplitude,Apostol, T.M. (1967) Calculus. 2nd edition. New York, NY, Wiley. Pp 334-335.{{MathWorld|id=HarmonicAdditionTheorem|title=Harmonic Addition Theorem}}

$$a\backslash cos\; x+b\backslash sin\; x=c\backslash cos(x+\backslash varphi)$$

where $c$ and $\backslash varphi$ are defined as so:

$$\backslash begin\{align\}$$

c &= \sgn(a) \sqrt{a^2 + b^2}, \\

\varphi &= {\arctan}\bigl({-b/a}\bigr),

\end{align}

given that $a\; \backslash neq\; 0.$

More generally, for arbitrary phase shifts, we have

$$a\; \backslash sin(x\; +\; \backslash theta\_a)\; +\; b\; \backslash sin(x\; +\; \backslash theta\_b)=\; c\; \backslash sin(x+\backslash varphi)$$

where $c$ and $\backslash varphi$ satisfy:

$$\backslash begin\{align\}$$

c^2 &= a^2 + b^2 + 2ab\cos \left(\theta_a - \theta_b \right) , \\

\tan \varphi &= \frac{a \sin \theta_a + b \sin \theta_b}{a \cos \theta_a + b \cos \theta_b}.

\end{align}

{{See also|phasor (sine waves)#Addition|label1=Phasor addition}}The general case reads

$$\backslash sum\_i\; a\_i\; \backslash sin(x\; +\; \backslash theta\_i)\; =\; a\; \backslash sin(x\; +\; \backslash theta),$$

where

$$a^2\; =\; \backslash sum\_\{i,j\}a\_i\; a\_j\; \backslash cos(\backslash theta\_i\; -\; \backslash theta\_j)$$

and

$$\backslash tan\backslash theta\; =\; \backslash frac\{\backslash sum\_i\; a\_i\; \backslash sin\backslash theta\_i\}\{\backslash sum\_i\; a\_i\; \backslash cos\backslash theta\_i\}.$$

These identities, named after Joseph Louis Lagrange, are:{{cite journal |first=Eddie |last=Ortiz Muñiz |date=Feb 1953 |volume=21 |number=2 |title=A Method for Deriving Various Formulas in Electrostatics and Electromagnetism Using Lagrange's Trigonometric Identities |journal=American Journal of Physics |page=140 | doi=10.1119/1.1933371 | bibcode=1953AmJPh..21..140M }}{{cite book |title=Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems | edition=illustrated |first1=Ravi P. |last1=Agarwal |first2=Donal |last2=O'Regan |publisher=Springer Science & Business Media |year=2008 |isbn=978-0-387-79146-3 |page=185 |url=https://books.google.com/books?id=jWvAfcNnphIC}} [https://books.google.com/books?id=jWvAfcNnphIC&pg=PA185 Extract of page 185]{{cite book |title=Handbook of Mathematical Formulas and Integrals |edition=4th |first1=Alan |last1=Jeffrey |first2=Hui-hui |last2=Dai |chapter=Section 2.4.1.6 |isbn=978-0-12-374288-9 |year=2008 |publisher=Academic Press}}

$$\backslash begin\{align\}$$

\sum_{k=0}^n \sin k\theta & = \frac{\cos \tfrac12\theta - \cos\left(\left(n + \tfrac12\right)\theta\right)}{2\sin\tfrac12\theta}\\[5pt]

\sum_{k=1}^n \cos k\theta & = \frac{-\sin \tfrac12\theta + \sin\left(\left(n + \tfrac12\right)\theta\right)}{2\sin\tfrac12\theta}

\end{align}

for $\backslash theta\; \backslash not\backslash equiv\; 0\; \backslash pmod\{2\backslash pi\}.$

A related function is the Dirichlet kernel:

$$D\_n(\backslash theta)\; =\; 1\; +\; 2\backslash sum\_\{k=1\}^n\; \backslash cos\; k\backslash theta$$

= \frac{\sin\left(\left(n + \tfrac12 \right)\theta\right)}{\sin \tfrac12 \theta}.

A similar identity is{{Cite journal |last1=Fay |first1=Temple H. |last2=Kloppers |first2=P. Hendrik |date=2001 |title=The Gibbs' phenomenon |url=http://dx.doi.org/10.1080/00207390117151 |journal=International Journal of Mathematical Education in Science and Technology |volume=32 |issue=1 |pages=73–89 |doi=10.1080/00207390117151}}

$$\backslash sum\_\{k=1\}^n\; \backslash cos\; (2k\; -1)\backslash alpha\; =\; \backslash frac\{\backslash sin\; (2n\; \backslash alpha)\}\{2\; \backslash sin\; \backslash alpha\}.$$

The proof is the following. By using the angle sum and difference identities,

$$\backslash sin\; (A\; +\; B)\; -\; \backslash sin\; (A\; -\; B)\; =\; 2\; \backslash cos\; A\; \backslash sin\; B.$$

Then let's examine the following formula,

$$2\; \backslash sin\; \backslash alpha\; \backslash sum\_\{k=1\}^n\; \backslash cos\; (2k\; -\; 1)\backslash alpha\; =\; 2\backslash sin\; \backslash alpha\; \backslash cos\; \backslash alpha\; +\; 2\; \backslash sin\; \backslash alpha\; \backslash cos\; 3\backslash alpha$$

+ 2 \sin \alpha \cos 5 \alpha + \ldots + 2 \sin \alpha \cos (2n - 1) \alpha

and this formula can be written by using the above identity,

$$\backslash begin\{align\}$$

& 2 \sin \alpha \sum_{k=1}^n \cos (2k - 1)\alpha \\

&\quad= \sum_{k=1}^n (\sin (2k \alpha) - \sin (2(k - 1)\alpha)) \\

&\quad= (\sin 2\alpha - \sin 0) + (\sin 4 \alpha - \sin 2 \alpha) + (\sin 6 \alpha - \sin 4 \alpha) + \ldots

+ (\sin (2n \alpha) - \sin (2(n - 1) \alpha)) \\

&\quad= \sin (2n \alpha).

\end{align}

So, dividing this formula with $2\; \backslash sin\; \backslash alpha$ completes the proof.

If $f(x)$ is given by the linear fractional transformation

$$f(x)\; =\; \backslash frac\{(\backslash cos\backslash alpha)x\; -\; \backslash sin\backslash alpha\}\{(\backslash sin\backslash alpha)x\; +\; \backslash cos\backslash alpha\},$$

and similarly

$$g(x)\; =\; \backslash frac\{(\backslash cos\backslash beta)x\; -\; \backslash sin\backslash beta\}\{(\backslash sin\backslash beta)x\; +\; \backslash cos\backslash beta\},$$

then

$$f\backslash big(g(x)\backslash big)\; =\; g\backslash big(f(x)\backslash big)$$

= \frac{\big(\cos(\alpha+\beta)\big)x - \sin(\alpha+\beta)}{\big(\sin(\alpha+\beta)\big)x + \cos(\alpha+\beta)}.

More tersely stated, if for all $\backslash alpha$ we let $f\_\{\backslash alpha\}$ be what we called $f$ above, then

$$f\_\backslash alpha\; \backslash circ\; f\_\backslash beta\; =\; f\_\{\backslash alpha+\backslash beta\}.$$

If $x$ is the slope of a line, then $f(x)$ is the slope of its rotation through an angle of $-\; \backslash alpha.$

{{Main|Euler's formula}}

Euler's formula states that, for any real number x:Abramowitz and Stegun, p. 74, 4.3.47

$$e^\{ix\}\; =\; \backslash cos\; x\; +\; i\backslash sin\; x,$$

where i is the imaginary unit. Substituting −x for x gives us:

$$e^\{-ix\}\; =\; \backslash cos(-x)\; +\; i\backslash sin(-x)\; =\; \backslash cos\; x\; -\; i\backslash sin\; x.$$

These two equations can be used to solve for cosine and sine in terms of the exponential function. Specifically,Abramowitz and Stegun, p. 71, 4.3.2Abramowitz and Stegun, p. 71, 4.3.1

$$\backslash cos\; x\; =\; \backslash frac\{e^\{ix\}\; +\; e^\{-ix\}\}\{2\}$$

$$\backslash sin\; x\; =\; \backslash frac\{e^\{ix\}\; -\; e^\{-ix\}\}\{2i\}$$

These formulae are useful for proving many other trigonometric identities. For example, that

{{math|1=e^i(θ+φ) = e^iθ e^iφ}} means that

{{block indent|em=1.5|text={{math|1=cos(θ + φ) + i sin(θ + φ) = (cos θ + i sin θ) (cos φ + i sin φ) = (cos θ cos φ − sin θ sin φ) + i (cos θ sin φ + sin θ cos φ)}}.}}

That the real part of the left hand side equals the real part of the right hand side is an angle addition formula for cosine. The equality of the imaginary parts gives an angle addition formula for sine.

The following table expresses the trigonometric functions and their inverses in terms of the exponential function and the complex logarithm.

Trigonometric functions may be deduced from hyperbolic functions with complex arguments. The formulae for the relations are shown below{{Cite book |last1=Hawkins |first1=Faith Mary |url=https://archive.org/details/isbn_356025055/mode/2up |title=Complex Numbers and Elementary Complex Functions |last2=Hawkins |first2=J. Q. |date=March 1, 1969 |publisher=MacDonald Technical & Scientific London |isbn=978-0356025056 |location=London |publication-date=1968 |pages=122 |language=english}}{{Cite book |last=Markushevich |first=A. I. |url=https://archive.org/details/markushevich-the-remarkable-sine-functions |title=The Remarkable Sine Function |publisher=American Elsevier Publishing Company, Inc. |year=1966 |isbn=978-1483256313 |location=New York |publication-date=1966 |pages=35–37, 81 |language=english}}.$$\backslash begin\{align\}$$

\sin x &= -i \sinh (ix) \\

\cos x &= \cosh (ix) \\

\tan x &= -i \tanh (i x) \\

\cot x &= i \coth (i x) \\

\sec x &= \operatorname{sech} (i x) \\

\csc x &= i \operatorname{csch} (i x) \\

\end{align}

When using a power series expansion to define trigonometric functions, the following identities are obtained:Abramowitz and Stegun, p. 74, 4.3.65–66

:$$\backslash sin\; x\; =\; x\; -\; \backslash frac\{x^3\}\{3!\}\; +\; \backslash frac\{x^5\}\{5!\}\; -\; \backslash frac\{x^7\}\{7!\}\; +\; \backslash cdots\; =\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash frac\{(-1)^nx^\{2n+1\}\}\{(2n+1)!\},$$$$\backslash cos\; x\; =\; 1\; -\; \backslash frac\{x^2\}\{2!\}\; +\; \backslash frac\{x^4\}\{4!\}\; -\; \backslash frac\{x^6\}\{6!\}\; +\; \backslash cdots\; =\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash frac\{(-1)^nx^\{2n\}\}\{(2n)!\}.$$

For applications to special functions, the following infinite product formulae for trigonometric functions are useful:Abramowitz and Stegun, p. 75, 4.3.89–90Abramowitz and Stegun, p. 85, 4.5.68–69

$$\backslash begin\{align\}$$

\sin x &= x \prod_{n = 1}^\infty\left(1 - \frac{x^2}{\pi^2 n^2}\right), &

\cos x &= \prod_{n = 1}^\infty\left(1 - \frac{x^2}{\pi^2\left(n - \frac{1}{2}\right)\!\vphantom)^2}\right), \\[10mu]

\sinh x &= x \prod_{n = 1}^\infty\left(1 + \frac{x^2}{\pi^2 n^2}\right), &

\cosh x &= \prod_{n = 1}^\infty\left(1 + \frac{x^2}{\pi^2\left(n - \frac{1}{2}\right)\!\vphantom)^2}\right).

\end{align}

{{Main|Inverse trigonometric functions}}

The following identities give the result of composing a trigonometric function with an inverse trigonometric function.{{harvnb|Abramowitz|Stegun|1972|loc=p. 73, 4.3.45}}

\begin{align}

\sin(\arcsin x) &=x

& \cos(\arcsin x) &=\sqrt{1-x^2}

& \tan(\arcsin x) &=\frac{x}{\sqrt{1 - x^2}}

\\

\sin(\arccos x) &=\sqrt{1-x^2}

& \cos(\arccos x) &=x

& \tan(\arccos x) &=\frac{\sqrt{1 - x^2}}{x}

\\

\sin(\arctan x) &=\frac{x}{\sqrt{1+x^2}}

& \cos(\arctan x) &=\frac{1}{\sqrt{1+x^2}}

& \tan(\arctan x) &=x

\\

\sin(\arccsc x) &=\frac{1}{x}

& \cos(\arccsc x) &=\frac{\sqrt{x^2 - 1}}{x}

& \tan(\arccsc x) &=\frac{1}{\sqrt{x^2 - 1}}

\\

\sin(\arcsec x) &=\frac{\sqrt{x^2 - 1}}{x}

& \cos(\arcsec x) &=\frac{1}{x}

& \tan(\arcsec x) &=\sqrt{x^2 - 1}

\\

\sin(\arccot x) &=\frac{1}{\sqrt{1+x^2}}

& \cos(\arccot x) &=\frac{x}{\sqrt{1+x^2}}

& \tan(\arccot x) &=\frac{1}{x}

\\

\end{align}

Taking the multiplicative inverse of both sides of the each equation above results in the equations for $\backslash csc\; =\; \backslash frac\{1\}\{\backslash sin\},\; \backslash ;\backslash sec\; =\; \backslash frac\{1\}\{\backslash cos\},\; \backslash text\{\; and\; \}\; \backslash cot\; =\; \backslash frac\{1\}\{\backslash tan\}.$

The right hand side of the formula above will always be flipped.

For example, the equation for $\backslash cot(\backslash arcsin\; x)$ is:

$$\backslash cot(\backslash arcsin\; x)\; =\; \backslash frac\{1\}\{\backslash tan(\backslash arcsin\; x)\}\; =\; \backslash frac\{1\}\{\backslash frac\{x\}\{\backslash sqrt\{1\; -\; x^2\}\}\}\; =\; \backslash frac\{\backslash sqrt\{1\; -\; x^2\}\}\{x\}$$

while the equations for $\backslash csc(\backslash arccos\; x)$ and $\backslash sec(\backslash arccos\; x)$ are:

$$\backslash csc(\backslash arccos\; x)\; =\; \backslash frac\{1\}\{\backslash sin(\backslash arccos\; x)\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{1-x^2\}\}\; \backslash qquad\; \backslash text\{\; and\; \}\backslash quad\; \backslash sec(\backslash arccos\; x)\; =\; \backslash frac\{1\}\{\backslash cos(\backslash arccos\; x)\}\; =\; \backslash frac\{1\}\{x\}.$$

The following identities are implied by the reflection identities. They hold whenever $x,\; r,\; s,\; -x,\; -r,\; \backslash text\{\; and\; \}\; -s$ are in the domains of the relevant functions.

$$\backslash begin\{alignat\}\{9\}$$

\frac{\pi}{2} ~&=~ \arcsin(x) &&+ \arccos(x) ~&&=~ \arctan(r) &&+ \arccot(r) ~&&=~ \arcsec(s) &&+ \arccsc(s) \\[0.4ex]

\pi ~&=~ \arccos(x) &&+ \arccos(-x) ~&&=~ \arccot(r) &&+ \arccot(-r) ~&&=~ \arcsec(s) &&+ \arcsec(-s) \\[0.4ex]

0 ~&=~ \arcsin(x) &&+ \arcsin(-x) ~&&=~ \arctan(r) &&+ \arctan(-r) ~&&=~ \arccsc(s) &&+ \arccsc(-s) \\[1.0ex]

\end{alignat}

Also,Wu, Rex H. "Proof Without Words: Euler's Arctangent Identity", Mathematics Magazine 77(3), June 2004, p. 189.

$$\backslash begin\{align\}$$

\arctan x + \arctan \dfrac{1}{x}

&= \begin{cases}

\frac{\pi}{2}, & \text{if } x > 0 \\

- \frac{\pi}{2}, & \text{if } x < 0

\end{cases} \\

\arccot x + \arccot \dfrac{1}{x}

&= \begin{cases}

\frac{\pi}{2}, & \text{if } x > 0 \\

\frac{3\pi}{2}, & \text{if } x < 0

\end{cases} \\

\end{align}

$$\backslash arccos\; \backslash frac\{1\}\{x\}\; =\; \backslash arcsec\; x\; \backslash qquad\; \backslash text\{\; and\; \}\; \backslash qquad\; \backslash arcsec\; \backslash frac\{1\}\{x\}\; =\; \backslash arccos\; x$$

$$\backslash arcsin\; \backslash frac\{1\}\{x\}\; =\; \backslash arccsc\; x\; \backslash qquad\; \backslash text\{\; and\; \}\; \backslash qquad\; \backslash arccsc\; \backslash frac\{1\}\{x\}\; =\; \backslash arcsin\; x$$

The arctangent function can be expanded as a series:{{citation | title = Algorithmic determination of a large integer in the two-term Machin-like formula for π | journal = Mathematics |author1=S. M. Abrarov|author2=R. K. Jagpal|author3=R. Siddiqui|author4=B. M. Quine | doi = 10.3390/math9172162 | year = 2021 | volume = 9 | issue = 17 | at = 2162| doi-access = free | arxiv = 2107.01027 }}

\arctan(nx) = \sum_{m = 1}^n \arctan\frac{x}{1 + (m - 1)mx^2}

In terms of the arctangent function we have

$$\backslash arctan\; \backslash frac\{1\}\{2\}\; =\; \backslash arctan\; \backslash frac\{1\}\{3\}\; +\; \backslash arctan\; \backslash frac\{1\}\{7\}.$$

The curious identity known as Morrie's law,

$$\backslash cos\; 20^\backslash circ\backslash cdot\backslash cos\; 40^\backslash circ\backslash cdot\backslash cos\; 80^\backslash circ\; =\; \backslash frac\{1\}\{8\},$$

is a special case of an identity that contains one variable:

$$\backslash prod\_\{j=0\}^\{k-1\}\backslash cos\backslash left(2^j\; x\backslash right)\; =\; \backslash frac\{\backslash sin\backslash left(2^k\; x\backslash right)\}\{2^k\backslash sin\; x\}.$$

Similarly,

$$\backslash sin\; 20^\backslash circ\backslash cdot\backslash sin\; 40^\backslash circ\backslash cdot\backslash sin\; 80^\backslash circ\; =\; \backslash frac\{\backslash sqrt\{3\}\}\{8\}$$

is a special case of an identity with $x\; =\; 20^\backslash circ$:

$$\backslash sin\; x\; \backslash cdot\; \backslash sin\; \backslash left(60^\backslash circ\; -\; x\backslash right)\; \backslash cdot\; \backslash sin\; \backslash left(60^\backslash circ\; +\; x\backslash right)\; =\; \backslash frac\{\backslash sin\; 3x\}\{4\}.$$

For the case $x\; =\; 15^\backslash circ$,

$$\backslash begin\{align\}$$

\sin 15^\circ\cdot\sin 45^\circ\cdot\sin 75^\circ &= \frac{\sqrt{2}}{8}, \\

\sin 15^\circ\cdot\sin 75^\circ &= \frac{1}{4}.

\end{align}

For the case $x\; =\; 10^\backslash circ$,

$$\backslash sin\; 10^\backslash circ\backslash cdot\backslash sin\; 50^\backslash circ\backslash cdot\backslash sin\; 70^\backslash circ\; =\; \backslash frac\{1\}\{8\}.$$

The same cosine identity is

$$\backslash cos\; x\; \backslash cdot\; \backslash cos\; \backslash left(60^\backslash circ\; -\; x\backslash right)\; \backslash cdot\; \backslash cos\; \backslash left(60^\backslash circ\; +\; x\backslash right)\; =\; \backslash frac\{\backslash cos\; 3x\}\{4\}.$$

Similarly,

$$\backslash begin\{align\}$$

\cos 10^\circ\cdot\cos 50^\circ\cdot\cos 70^\circ &= \frac{\sqrt{3}}{8}, \\

\cos 15^\circ\cdot\cos 45^\circ\cdot\cos 75^\circ &= \frac{\sqrt{2}}{8}, \\

\cos 15^\circ\cdot\cos 75^\circ &= \frac{1}{4}.

\end{align}

Similarly,

$$\backslash begin\{align\}$$

\tan 50^\circ\cdot\tan 60^\circ\cdot\tan 70^\circ &= \tan 80^\circ, \\

\tan 40^\circ\cdot\tan 30^\circ\cdot\tan 20^\circ &= \tan 10^\circ.

\end{align}

The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):

$$\backslash cos\; 24^\backslash circ\; +\; \backslash cos\; 48^\backslash circ\; +\; \backslash cos\; 96^\backslash circ\; +\; \backslash cos\; 168^\backslash circ\; =\; \backslash frac\{1\}\{2\}.$$

Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:

\cos \frac{2\pi}{21} +

\cos\left(2\cdot\frac{2\pi}{21}\right) +

\cos\left(4\cdot\frac{2\pi}{21}\right) +

\cos\left( 5\cdot\frac{2\pi}{21}\right) +

\cos\left( 8\cdot\frac{2\pi}{21}\right) +

\cos\left(10\cdot\frac{2\pi}{21}\right)

= \frac{1}{2}.

The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than {{sfrac|21|2}} that are relatively prime to (or have no prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.

Other cosine identities include:{{cite journal|last=Humble |first=Steve |title=Grandma's identity |journal=Mathematical Gazette |volume=88 |date=Nov 2004 |pages=524–525 |doi=10.1017/s0025557200176223|s2cid=125105552 }}

$$\backslash begin\{align\}$$

2\cos \frac{\pi}{3} &= 1, \\

2\cos \frac{\pi}{5} \times 2\cos \frac{2\pi}{5} &= 1, \\

2\cos \frac{\pi}{7} \times 2\cos \frac{2\pi}{7}\times 2\cos \frac{3\pi}{7} &= 1,

\end{align}

and so forth for all odd numbers, and hence

$$\backslash cos\; \backslash frac\{\backslash pi\}\{3\}+\backslash cos\; \backslash frac\{\backslash pi\}\{5\}\; \backslash times\; \backslash cos\; \backslash frac\{2\backslash pi\}\{5\}\; +\; \backslash cos\; \backslash frac\{\backslash pi\}\{7\}\; \backslash times\; \backslash cos\; \backslash frac\{2\backslash pi\}\{7\}\; \backslash times\; \backslash cos\; \backslash frac\{3\backslash pi\}\{7\}\; +\; \backslash dots\; =\; 1.$$

Many of those curious identities stem from more general facts like the following:{{MathWorld|id=Sine|title=Sine}}

$$\backslash prod\_\{k=1\}^\{n-1\}\; \backslash sin\backslash frac\{k\backslash pi\}\{n\}\; =\; \backslash frac\{n\}\{2^\{n-1\}\}$$

and

$$\backslash prod\_\{k=1\}^\{n-1\}\; \backslash cos\backslash frac\{k\backslash pi\}\{n\}\; =\; \backslash frac\{\backslash sin\backslash frac\{\backslash pi\; n\}\{2\}\}\{2^\{n-1\}\}.$$

Combining these gives us

$$\backslash prod\_\{k=1\}^\{n-1\}\; \backslash tan\backslash frac\{k\backslash pi\}\{n\}\; =\; \backslash frac\{n\}\{\backslash sin\backslash frac\{\backslash pi\; n\}\{2\}\}$$

If {{mvar|n}} is an odd number ($n\; =\; 2\; m\; +\; 1$) we can make use of the symmetries to get

$$\backslash prod\_\{k=1\}^\{m\}\; \backslash tan\backslash frac\{k\backslash pi\}\{2m+1\}\; =\; \backslash sqrt\{2m+1\}$$

The transfer function of the Butterworth low pass filter can be expressed in terms of polynomial and poles. By setting the frequency as the cutoff frequency, the following identity can be proved:

$$\backslash prod\_\{k=1\}^n\; \backslash sin\backslash frac\{\backslash left(2k\; -\; 1\backslash right)\backslash pi\}\{4n\}\; =\; \backslash prod\_\{k=1\}^\{n\}\; \backslash cos\backslash frac\{\backslash left(2k-1\backslash right)\backslash pi\}\{4n\}\; =\; \backslash frac\{\backslash sqrt\{2\}\}\{2^n\}$$

An efficient way to compute {{pi}} to a large number of digits is based on the following identity without variables, due to Machin. This is known as a Machin-like formula:

$$\backslash frac\{\backslash pi\}\{4\}\; =\; 4\; \backslash arctan\backslash frac\{1\}\{5\}\; -\; \backslash arctan\backslash frac\{1\}\{239\}$$

or, alternatively, by using an identity of Leonhard Euler:

$$\backslash frac\{\backslash pi\}\{4\}\; =\; 5\; \backslash arctan\backslash frac\{1\}\{7\}\; +\; 2\; \backslash arctan\backslash frac\{3\}\{79\}$$

or by using Pythagorean triples:

$$\backslash pi\; =\; \backslash arccos\backslash frac\{4\}\{5\}\; +\; \backslash arccos\backslash frac\{5\}\{13\}\; +\; \backslash arccos\backslash frac\{16\}\{65\}\; =\; \backslash arcsin\backslash frac\{3\}\{5\}\; +\; \backslash arcsin\backslash frac\{12\}\{13\}\; +\; \backslash arcsin\backslash frac\{63\}\{65\}.$$

Others include:Harris, Edward M. "Sums of Arctangents", in Roger B. Nelson, Proofs Without Words (1993, Mathematical Association of America), p. 39.

$$\backslash frac\{\backslash pi\}\{4\}\; =\; \backslash arctan\backslash frac\{1\}\{2\}\; +\; \backslash arctan\backslash frac\{1\}\{3\},$$

$$\backslash pi\; =\; \backslash arctan\; 1\; +\; \backslash arctan\; 2\; +\; \backslash arctan\; 3,$$

$$\backslash frac\{\backslash pi\}\{4\}\; =\; 2\backslash arctan\; \backslash frac\{1\}\{3\}\; +\; \backslash arctan\; \backslash frac\{1\}\{7\}.$$

Generally, for numbers {{math|t₁, ..., t_n−1 ∈ (−1, 1)}} for which {{math|1=θ_n = Σ{{su|b=k=1|p=n−1}} arctan t_k ∈ (π/4, 3π/4)}}, let {{math|1=t_n = tan(π/2 − θ_n) = cot θ_n}}. This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are {{math|t₁, ..., t_n−1}} and its value will be in {{math|(−1, 1)}}. In particular, the computed {{math|t_n}} will be rational whenever all the {{math|t₁, ..., t_n−1}} values are rational. With these values,

$$\backslash begin\{align\}$$

\frac{\pi}{2} & = \sum_{k=1}^n \arctan(t_k) \\

\pi & = \sum_{k=1}^n \sgn(t_k) \arccos\left(\frac{1 - t_k^2}{1 + t_k^2}\right) \\

\pi & = \sum_{k=1}^n \arcsin\left(\frac{2t_k}{1 + t_k^2}\right) \\

\pi & = \sum_{k=1}^n \arctan\left(\frac{2t_k}{1 - t_k^2}\right)\,,

\end{align}

where in all but the first expression, we have used tangent half-angle formulae. The first two formulae work even if one or more of the {{math|t_k}} values is not within {{math|(−1, 1)}}. Note that if {{math|1=t = p/q}} is rational, then the {{math|(2t, 1 − t², 1 + t²)}} values in the above formulae are proportional to the Pythagorean triple {{math|(2pq, q² − p², q² + p²)}}.

For example, for {{math|1=n = 3}} terms,

$$\backslash frac\{\backslash pi\}\{2\}\; =\; \backslash arctan\backslash left(\backslash frac\{a\}\{b\}\backslash right)\; +\; \backslash arctan\backslash left(\backslash frac\{c\}\{d\}\backslash right)\; +\; \backslash arctan\backslash left(\backslash frac\{bd\; -\; ac\}\{ad\; +\; bc\}\backslash right)$$

for any {{math|a, b, c, d > 0}}.

Euclid showed in Book XIII, Proposition 10 of his Elements that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says:

$$\backslash sin^2\; 18^\backslash circ\; +\; \backslash sin^2\; 30^\backslash circ\; =\; \backslash sin^2\; 36^\backslash circ.$$

Ptolemy used this proposition to compute some angles in his table of chords in Book I, chapter 11 of Almagest.

These identities involve a trigonometric function of a trigonometric function:Milton Abramowitz and Irene Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1972, formulae 9.1.42–9.1.45

: $\backslash cos(t\; \backslash sin\; x)\; =\; J\_0(t)\; +\; 2\; \backslash sum\_\{k=1\}^\backslash infty\; J\_\{2k\}(t)\; \backslash cos(2kx)$

: $\backslash sin(t\; \backslash sin\; x)\; =\; 2\; \backslash sum\_\{k=0\}^\backslash infty\; J\_\{2k+1\}(t)\; \backslash sin\backslash big((2k+1)x\backslash big)$

: $\backslash cos(t\; \backslash cos\; x)\; =\; J\_0(t)\; +\; 2\; \backslash sum\_\{k=1\}^\backslash infty\; (-1)^kJ\_\{2k\}(t)\; \backslash cos(2kx)$

: $\backslash sin(t\; \backslash cos\; x)\; =\; 2\; \backslash sum\_\{k=0\}^\backslash infty(-1)^k\; J\_\{2k+1\}(t)\; \backslash cos\backslash big((2k+1)x\backslash big)$

where {{mvar|J_i}} are Bessel functions.

A conditional trigonometric identity is a trigonometric identity that holds if specified conditions on the arguments to the trigonometric functions are satisfied.Er. K. C. Joshi, Krishna's IIT MATHEMATIKA. Krishna Prakashan Media. Meerut, India. page 636. The following formulae apply to arbitrary plane triangles and follow from $\backslash alpha\; +\; \backslash beta\; +\; \backslash gamma\; =\; 180^\{\backslash circ\},$ as long as the functions occurring in the formulae are well-defined (the latter applies only to the formulae in which tangents and cotangents occur).Cagnoli, Antonio (1808), Trigonométrie rectiligne et sphérique, p. 27.

$$\backslash begin\{align\}$$

\tan \alpha + \tan \beta + \tan \gamma &= \tan \alpha \tan \beta \tan \gamma \\

1 &= \cot \beta \cot \gamma + \cot \gamma \cot \alpha + \cot \alpha \cot \beta \\

\cot\left(\frac{\alpha}{2}\right) + \cot\left(\frac{\beta}{2}\right) + \cot\left(\frac{\gamma}{2}\right) &= \cot\left(\frac{\alpha}{2}\right) \cot \left(\frac{\beta}{2}\right) \cot\left(\frac{\gamma}{2}\right) \\

1 &= \tan\left(\frac{\beta}{2}\right)\tan\left(\frac{\gamma}{2}\right) + \tan\left(\frac{\gamma}{2}\right)\tan\left(\frac{\alpha}{2}\right) + \tan\left(\frac{\alpha}{2}\right)\tan\left(\frac{\beta}{2}\right) \\

\sin \alpha + \sin \beta + \sin \gamma &= 4\cos\left(\frac{\alpha}{2}\right)\cos\left(\frac{\beta}{2}\right)\cos\left(\frac{\gamma}{2}\right) \\

-\sin \alpha + \sin \beta + \sin \gamma &= 4\cos\left(\frac{\alpha}{2}\right)\sin\left(\frac{\beta}{2}\right)\sin\left(\frac{\gamma}{2}\right) \\

\cos \alpha + \cos \beta + \cos \gamma &= 4\sin\left(\frac{\alpha}{2}\right)\sin\left(\frac{\beta}{2}\right)\sin \left(\frac{\gamma}{2}\right) + 1 \\

-\cos \alpha + \cos \beta + \cos \gamma &= 4\sin\left(\frac{\alpha}{2}\right)\cos\left(\frac{\beta}{2}\right)\cos \left(\frac{\gamma}{2}\right) - 1 \\

\sin (2\alpha) + \sin (2\beta) + \sin (2\gamma) &= 4\sin \alpha \sin \beta \sin \gamma \\

-\sin (2\alpha) + \sin (2\beta) + \sin (2\gamma) &= 4\sin \alpha \cos \beta \cos \gamma \\

\cos (2\alpha) + \cos (2\beta) + \cos (2\gamma) &= -4\cos \alpha \cos \beta \cos \gamma - 1 \\

-\cos (2\alpha) + \cos (2\beta) + \cos (2\gamma) &= -4\cos \alpha \sin \beta \sin \gamma + 1 \\

\sin^2\alpha + \sin^2\beta + \sin^2\gamma &= 2 \cos \alpha \cos \beta \cos \gamma + 2 \\

-\sin^2\alpha + \sin^2\beta + \sin^2\gamma &= 2 \cos \alpha \sin \beta \sin \gamma \\

\cos^2\alpha + \cos^2\beta + \cos^2\gamma &= -2 \cos \alpha \cos \beta \cos \gamma + 1 \\

-\cos^2\alpha + \cos^2\beta + \cos^2\gamma &= -2 \cos \alpha \sin \beta \sin \gamma + 1 \\

\sin^2 (2\alpha) + \sin^2 (2\beta) + \sin^2 (2\gamma) &= -2\cos (2\alpha) \cos (2\beta) \cos (2\gamma)+2 \\

\cos^2 (2\alpha) + \cos^2 (2\beta) + \cos^2 (2\gamma) &= 2\cos (2\alpha) \,\cos (2\beta) \,\cos (2\gamma) + 1 \\

1 &= \sin^2 \left(\frac{\alpha}{2}\right) + \sin^2 \left(\frac{\beta}{2}\right) + \sin^2 \left(\frac{\gamma}{2}\right) + 2\sin \left(\frac{\alpha}{2}\right) \,\sin \left(\frac{\beta}{2}\right) \,\sin \left(\frac{\gamma}{2}\right)

\end{align}

{{Main|Versine|Exsecant}}

The versine, coversine, haversine, and exsecant were used in navigation. For example, the haversine formula was used to calculate the distance between two points on a sphere. They are rarely used today.

==Miscellaneous==

{{Main|Dirichlet kernel}}

The Dirichlet kernel {{math|D_n(x)}} is the function occurring on both sides of the next identity:

$$1\; +\; 2\backslash cos\; x\; +\; 2\backslash cos(2x)\; +\; 2\backslash cos(3x)\; +\; \backslash cdots\; +\; 2\backslash cos(nx)\; =\; \backslash frac\{\backslash sin\backslash left(\backslash left(n\; +\; \backslash frac\{1\}\{2\}\backslash right)x\backslash right)\; \}\{\backslash sin\backslash left(\backslash frac\{1\}\{2\}x\backslash right)\}.$$

The convolution of any integrable function of period $2\; \backslash pi$ with the Dirichlet kernel coincides with the function's $n$th-degree Fourier approximation. The same holds for any measure or generalized function.

{{Main|Tangent half-angle substitution}}

If we set $$t\; =\; \backslash tan\backslash frac\; x\; 2,$$ thenAbramowitz and Stegun, p. 72, 4.3.23

$$\backslash sin\; x\; =\; \backslash frac\{2t\}\{1\; +\; t^2\};\backslash qquad\; \backslash cos\; x\; =\; \backslash frac\{1\; -\; t^2\}\{1\; +\; t^2\};\backslash qquad\; e^\{i\; x\}\; =\; \backslash frac\{1\; +\; i\; t\}\{1\; -\; i\; t\};\; \backslash qquad\; dx\; =\; \backslash frac\{2\backslash ,dt\}\{1+t^2\},$$

where $e^\{i\; x\}\; =\; \backslash cos\; x\; +\; i\; \backslash sin\; x,$ sometimes abbreviated to {{math|cis x}}.

When this substitution of $t$ for {{math|tan {{sfrac|x|2}}}} is used in calculus, it follows that $\backslash sin\; x$ is replaced by {{math|{{sfrac|2t|1 + t²}}}}, $\backslash cos\; x$ is replaced by {{math|{{sfrac|1 − t²|1 + t²}}}} and the differential {{math|dx}} is replaced by {{math|{{sfrac|2 dt|1 + t²}}}}. Thereby one converts rational functions of $\backslash sin\; x$ and $\backslash cos\; x$ to rational functions of $t$ in order to find their antiderivatives.

{{See also|Viète's formula|Sinc function}}

$$\backslash cos\backslash frac\{\backslash theta\}\{2\}\; \backslash cdot\; \backslash cos\; \backslash frac\{\backslash theta\}\{4\}$$

\cdot \cos \frac{\theta}{8} \cdots = \prod_{n=1}^\infty \cos \frac{\theta}{2^n}

= \frac{\sin \theta}{\theta} = \operatorname{sinc} \theta.

{{div col|colwidth=30em}}

• Aristarchus's inequality
• Derivatives of trigonometric functions
• Exact trigonometric values (values of sine and cosine expressed in surds)
• Exsecant
• Half-side formula
• Hyperbolic function
• Laws for solution of triangles:
• Law of cosines
• Spherical law of cosines
• Law of sines
• Law of tangents
• Law of cotangents
• Mollweide's formula
• List of integrals of trigonometric functions
• Mnemonics in trigonometry
• Pentagramma mirificum
• Proofs of trigonometric identities
• Prosthaphaeresis
• Pythagorean theorem
• Tangent half-angle formula
• Trigonometric number
• Trigonometry
• Uses of trigonometry
• Versine and haversine

{{div col end}}

{{reflist|30em}}

{{Refbegin}}

• {{Cite book|editor1-last=Abramowitz|editor1-first=Milton|editor1-link=Milton Abramowitz|editor2-last=Stegun|editor2-first=Irene A.|editor2-link=Irene Stegun|title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables|publisher=Dover Publications|location=New York|isbn=978-0-486-61272-0|year=1972|url=https://archive.org/details/handbookofmathe000abra }}
• {{ citation|last1 = Nielsen|first1 = Kaj L.|title = Logarithmic and Trigonometric Tables to Five Places|edition = 2nd|location = New York|publisher = Barnes & Noble|year = 1966|lccn = 61-9103 }}
• {{citation|editor-first=Samuel M.|editor-last=Selby|title=Standard Mathematical Tables|publisher=The Chemical Rubber Co.|year=1970|edition=18th}}

{{Refend}}

• [http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html Values of sin and cos, expressed in surds, for integer multiples of 3° and of {{sfrac|5|5|8}}°], and for the same angles [http://www.jdawiseman.com/papers/easymath/surds_csc_sec.html csc and sec] and [http://www.jdawiseman.com/papers/easymath/surds_tan.html tan]

{{DEFAULTSORT:Trigonometric identities}}

Category:Mathematical identities

Identities

Category:Mathematics-related lists