In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common trick involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

Notation

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The following notations hold for all six trigonometric functions: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). For brevity, only the sine case is given in the table.

Notation	Reading	Description	Definition
sin2(x)	"sine squared * x"	the square of sine; sine to the second power	sin2(x) = (sin(x))2
arcsin(x)	"arcsine * x"	the inverse function for sine	arcsin(x) = y  if and only if  sin(y) = x and $-\{\backslash pi\; \backslash over\; 2\}\; \backslash le\; y\; \backslash le\; \{\backslash pi\; \backslash over\; 2\}$
(sin(x))−1	"sine * x, to the power of minus-one"	the reciprocal of sine; the multiplicative inverse of sine	(sin(x))−1 = 1 / sin(x)

arcsin(x) can also be written sin⁻¹(x); this must not be confused with (sin(x))⁻¹.

Definitions

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$\backslash cos(x)\; =\; \backslash sin\backslash left(\; x\; +\; \{\backslash pi\; \backslash over\; 2\}\; \backslash right)$

$\backslash tan\; (x)\; =\; \backslash frac\; \{\backslash sin\; (x)\}\; \{\backslash cos(x)\}\; \backslash qquad\; \backslash operatorname\{cot\}(x)\; =\; \backslash frac\; \{\backslash cos\; (x)\}\; \{\backslash sin(x)\}\; =\; \backslash frac\{1\}\; \{\backslash tan(x)\}$

$\backslash operatorname\{sec\}(x)\; =\; \backslash frac\{1\}\; \{\backslash cos(x)\}\; \backslash qquad\; \backslash operatorname\{csc\}(x)\; =\; \backslash frac\{1\}\; \{\backslash sin(x)\}$

For more information, including definitions based on the sides of a right triangle, see trigonometric function.

Periodicity, symmetry, and shifts

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These are most easily shown from the unit circle:

Periodicity

The sine, cosine, secant, and cosecant functions have period 2π (a full circle):

$\backslash sin(x)\; =\; \backslash sin(x\; +\; 2k\backslash pi)\; \backslash ,$

$\backslash cos(x)\; =\; \backslash cos(x\; +\; 2k\backslash pi)\; \backslash ,$

$\backslash sec(x)\; =\; \backslash sec(x\; +\; 2k\backslash pi)\; \backslash ,$

$\backslash csc(x)\; =\; \backslash csc(x\; +\; 2k\backslash pi)\; \backslash ,$

The tangent and cotangent functions have period π (a half-circle):

$\backslash tan(x)\; =\; \backslash tan(x\; +\; k\backslash pi)\; \backslash ,$

$\backslash cot(x)\; =\; \backslash cot(x\; +\; k\backslash pi)\; \backslash ,$

Symmetry

The sine, cosecant, tangent, and cotangent functions are odd:

$\backslash sin(-x)\; =\; -\backslash sin(x)\; \backslash ,$

$\backslash csc(-x)\; =\; -\backslash csc(x)\; \backslash ,$

$\backslash tan(-x)\; =\; -\backslash tan(x)\; \backslash ,$

$\backslash cot(-x)\; =\; -\backslash cot(x)\; \backslash ,$

The cosine and secant functions are even:

$\backslash cos(-x)\; =\backslash ;\; \backslash cos(x)\; \backslash ,$

$\backslash sec(-x)\; =\backslash ;\; \backslash sec(x)\; \backslash ,$

Another symmetry affects all six of these functions:

$$

\left.\begin{matrix} \sin\left({\pi \over 2} - x\right) = \cos(x) \\ \\ \cos\left({\pi \over 2} - x\right) = \sin(x) \end{matrix}\right\}\mbox{These two are equivalent.}

$$

\left.\begin{matrix} \tan\left({\pi \over 2} - x\right) = \cot(x) \\ \\ \cot\left({\pi \over 2} - x\right) = \tan(x) \end{matrix}\right\}\mbox{These two are equivalent.}

$$

\left.\begin{matrix} \sec\left({\pi \over 2} - x\right) = \csc(x) \\ \\ \csc\left({\pi \over 2} - x\right) = \sec(x) \end{matrix}\right\}\mbox{These two are equivalent.}

Shifts

Among the simplest shifts (other than shifts by the period of each of these periodic functions) are shifts by π/2:

$$

\left.\begin{matrix} \sin\left( x + {\pi \over 2} \right) = \cos(x) \\ \\ \cos\left( x + {\pi \over 2} \right) = -\sin(x) \end{matrix}\right\}\mbox{one with a minus sign; one without}

$$

\left.\begin{matrix} \tan\left( x + {\pi \over 2} \right) = -\cot(x) \\ \\ \cot\left( x + {\pi \over 2} \right) = -\tan(x) \end{matrix}\right\}\mbox{both with minus signs}

$$

\left.\begin{matrix} \sec\left( x + {\pi \over 2} \right) = -\csc(x) \\ \\ \csc\left( x + {\pi \over 2} \right) = \sec(x) \end{matrix}\right\}\mbox{one with a minus sign; one without}

For some purposes it is important to know that any linear combination of sine waves of the same period but different phase shifts is also a sine wave with the same period, but a different phase shift. In other words, we have

$a\backslash sin\; x+b\backslash cos\; x=\backslash sqrt\{a^2+b^2\}\backslash cdot\backslash sin(x+\backslash varphi)\backslash ,$

where

$$

\varphi= \left\{ \begin{matrix} {\rm arctan}(b/a),&&\mbox{if }a\ge0; \; \\ \pi-{\rm arctan}(b/a),&&\mbox{if }a<0. \; \end{matrix} \right. \;

Pythagorean identities

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These identities are based on the Pythagorean theorem. The first is sometimes simply called the Pythagorean trigonometric identity.

$\backslash sin^2(x)\; +\; \backslash cos^2(x)\; =\; 1\; \backslash ;$

$\backslash tan^2(x)\; +\; 1\; =\; \backslash sec^2(x)\; \backslash ;$

$1\; +\; \backslash cot^2(x)\; =\; \backslash csc^2(x)\; \backslash ;$

Note that the second equation is obtained from the first by dividing both sides by cos²(x). To get the third equation, divide the first by sin²(x) instead.

Angle sum and difference identities

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These are also known as the addition and subtraction theorems or formulae. The quickest way to prove these is Euler's formula. The tangent formula follows from the other two. A geometric proof of the sin(x + y) identity is given at the end of this article.

$\backslash sin(x\; \backslash pm\; y)\; =\; \backslash sin(x)\; \backslash cos(y)\; \backslash pm\; \backslash cos(x)\; \backslash sin(y)\backslash ,$

$\backslash cos(x\; \backslash pm\; y)\; =\; \backslash cos(x)\; \backslash cos(y)\; \backslash mp\; \backslash sin(x)\; \backslash sin(y)\backslash ,$

(When "+" is on the left side, then "−" is on the right, and vice versa.)

$\backslash tan(x\; \backslash pm\; y)\; =\; \backslash frac\{\backslash tan(x)\; \backslash pm\; \backslash tan(y)\}\{1\; \backslash mp\; \backslash tan(x)\backslash tan(y)\}$

$\{\backslash rm\; c\backslash dot\{\backslash imath\}\; s\}(x+y)=\{\backslash rm\; c\backslash dot\{\backslash imath\}\; s\}(x)\backslash ,\{\backslash rm\; c\backslash dot\{\backslash imath\}\; s\}(y)$

$\{\backslash rm\; c\backslash dot\{\backslash imath\}\; s\}(x-y)=$

$\backslash sin\backslash left(\backslash frac\{x\}\{2\}\backslash right)\; =\; \backslash pm\backslash ,\; \backslash sqrt\{\backslash frac\{1\; -\; \backslash cos(x)\}\{2\}\}$

These may also be called the half-angle formulae. Then

$\backslash tan\backslash left(\backslash frac\{x\}\{2\}\backslash right)\; =\; \{\backslash sin\; (x/2)\; \backslash over\; \backslash cos\; (x/2)\}\; =\; \backslash pm\backslash ,\; \backslash sqrt\{1\; -\; \backslash cos\; x\; \backslash over\; 1\; +\; \backslash cos\; x\}.\; \backslash qquad\; \backslash qquad\; (1)$

Multiply both numerator and denominator inside the radical by 1 + cos x, then simplify (using a Pythagorean identity):

$\backslash tan\backslash left(\backslash frac\{x\}\{2\}\backslash right)\; =\; \backslash pm\backslash ,\; \backslash sqrt\{(1\; -\; \backslash cos\; x)\; (1\; +\; \backslash cos\; x)\; \backslash over\; (1\; +\; \backslash cos\; x)\; (1\; +\; \backslash cos\; x)\}\; =\; \backslash pm\backslash ,\; \backslash sqrt\{1\; -\; \backslash cos^2\; x\; \backslash over\; (1\; +\; \backslash cos\; x)^2\}$

$=\; \{\backslash sin\; x\; \backslash over\; 1\; +\; \backslash cos\; x\}.$

Likewise, multiplying both numerator and denominator inside the radical — in equation (1) — by
1 − cos x, then simplifying:

$\backslash tan\backslash left(\backslash frac\{x\}\{2\}\backslash right)\; =\; \backslash pm\backslash ,\; \backslash sqrt\{(1\; -\; \backslash cos\; x)\; (1\; -\; \backslash cos\; x)\; \backslash over\; (1\; +\; \backslash cos\; x)\; (1\; -\; \backslash cos\; x)\}\; =\; \backslash pm\backslash ,\; \backslash sqrt\{(1\; -\; \backslash cos\; x)^2\; \backslash over\; (1\; -\; \backslash cos^2\; x)\}$

$=\; \{1\; -\; \backslash cos\; x\; \backslash over\; \backslash sin\; x\}.$

Thus, the pair of half-angle formulae for the tangent are:

$\backslash tan\backslash left(\backslash frac\{x\}\{2\}\backslash right)\; =\; \backslash frac\{\backslash sin(x)\}\{1\; +\; \backslash cos(x)\}\; =\; \backslash frac\{1-\backslash cos(x)\}\{\backslash sin(x)\}.$

If we set

$t\; =\; \backslash tan\backslash left(\backslash frac\{x\}\{2\}\backslash right),$

then

$\backslash sin(x)\; =\; \backslash frac\{2t\}\{1\; +\; t^2\}$

and

$\backslash cos(x)\; =\; \backslash frac\{1\; -\; t^2\}\{1\; +\; t^2\}$

and

$e^\{i\; x\}\; =\; \backslash frac\{1\; +\; i\; t\}\{1\; -\; i\; t\}.$

This substitution of t for tan(x/2), with the consequent replacement of sin(x) by 2t/(1 + t²) and cos(x) by (1 − t²)/(1 + t²) is useful in calculus for converting rational functions in sin(x) and cos(x) to functions of t in order to find their antiderivatives. For more information see tangent half-angle formula.

Product-to-sum identities

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These can be proven by expanding their right-hand sides using the angle addition theorems.

$\backslash cos\backslash left\; (x\backslash right\; )\; \backslash cos\backslash left\; (y\backslash right\; )\; =\; \{\backslash cos\backslash left\; (x\; -\; y\backslash right\; )\; +\; \backslash cos\backslash left\; (x\; +\; y\backslash right\; )\; \backslash over\; 2\}\; \backslash ;$

$\backslash sin\backslash left\; (x\backslash right\; )\; \backslash sin\backslash left\; (y\backslash right\; )\; =\; \{\backslash cos\backslash left\; (x\; -\; y\backslash right\; )\; -\; \backslash cos\backslash left\; (x\; +\; y\backslash right\; )\; \backslash over\; 2\}\; \backslash ;$

$\backslash sin\backslash left\; (x\backslash right\; )\; \backslash cos\backslash left\; (y\backslash right\; )\; =\; \{\backslash sin\backslash left\; (x\; -\; y\backslash right\; )\; +\; \backslash sin\backslash left\; (x\; +\; y\backslash right\; )\; \backslash over\; 2\}\; \backslash ;$

Sum-to-product identities

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Replace x by (x + y) / 2 and y by (x – y) / 2 in the product-to-sum formulae.

$\backslash cos(x)\; +\; \backslash cos(y)\; =\; 2\; \backslash cos\backslash left(\; \backslash frac\{x\; +\; y\}\{2\}\; \backslash right)\; \backslash cos\backslash left(\; \backslash frac\{x\; -\; y\}\{2\}\; \backslash right)\; \backslash ;$

$\backslash sin(x)\; +\; \backslash sin(y)\; =\; 2\; \backslash sin\backslash left(\; \backslash frac\{x\; +\; y\}\{2\}\; \backslash right)\; \backslash cos\backslash left(\; \backslash frac\{x\; -\; y\}\{2\}\; \backslash right)\; \backslash ;$

$\backslash cos(x)\; -\; \backslash cos(y)\; =\; -2\; \backslash sin\backslash left(\; \{x\; +\; y\; \backslash over\; 2\}\backslash right)\; \backslash sin\backslash left(\{x\; -\; y\; \backslash over\; 2\}\backslash right)\; \backslash ;$

$\backslash sin(x)\; -\; \backslash sin(y)\; =\; 2\; \backslash cos\backslash left(\{x\; +\; y\backslash over\; 2\}\backslash right)\; \backslash sin\backslash left(\{x\; -\; y\backslash over\; 2\}\backslash right)\; \backslash ;$

If x, y, and z are the three angles of any triangle, or in other words

$\backslash mbox\{if\; \}x\; +\; y\; +\; z\; =\; \backslash pi\; =\; \backslash mbox\{half\; circle,\}\backslash ,$

$\backslash mbox\{then\; \}\backslash tan(x)\; +\; \backslash tan(y)\; +\; \backslash tan(z)\; =\; \backslash tan(x)\backslash tan(y)\backslash tan(z).\backslash ,$

(If any of x, y, z is a right angle, one should take both sides to be ∞. This is neither +∞ nor −∞; for present purposes it makes sense to add just one point at infinity to the real line, that is approached by tan(θ) as tan(θ) either increases through positive values or decreases through negative values. This is a one-point compactification of the real line.)

Other sums of trigonometric functions

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For any a and b:

$a\; \backslash cos(x)\; +\; b\; \backslash sin(x)\; =\; \backslash sqrt\{\; a^2\; +\; b^2\; \}\; \backslash cos(x\; -\; \backslash arctan(b,\; a))\; \backslash ;$

where arctan(y, x) is the generalization of arctan(y/x) which covers the entire circular range (see also the account of this same identity in "symmetry, periodicity, and shifts" above for this generalization of arctan).

$\backslash tan(x)\; +\; \backslash sec(x)\; =\; \backslash tan\backslash left(\{x\; \backslash over\; 2\}\; +\; \{\backslash pi\; \backslash over\; 4\}\backslash right).$

The above identity is sometimes convenient to know when thinking about the Gudermanian function.

If $x$, $y$, and $z$ are the three angles of any triangle, i.e. $x\; +\; y\; +\; z\; =\; \backslash pi$ then,

$\backslash cot(x)\backslash cot(y)\; +\; \backslash cot(y)\backslash cot(z)\; +\; \backslash cot(z)\backslash cot(x)\; =\; 1.\backslash ,$

Inverse trigonometric functions

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$\backslash arcsin(x)+\backslash arccos(x)=\backslash pi/2\backslash ;$

$\backslash arctan(x)+\backslash arccot(x)=\backslash pi/2.\backslash ;$

$\backslash sin(\backslash arccos(x))=\backslash sqrt\{1-x^2\}\; \backslash ,$

$\backslash cos(\backslash arcsin(x))=\backslash sqrt\{1-x^2\}\; \backslash ,$

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

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

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

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

Every trigonometric function can be related directly to every other trigonometric function. Such relations can be expressed by means of inverse trigonometric functions as follows: let φ and ψ represent a pair of trigonometric functions, and let arcψ be the inverse of ψ, such that ψ(arcψ(x))=x. Then φ(arcψ(x)) can be expressed as an algebraic formula in terms of x. Such formulae are shown in the table below: φ can be made equal to the head of one of the rows, and ψ can be equated to the head of a column:

Table of conversion formulae

φ \ ψ	sin	cos	tan	csc	sec	cot
sin	$x\backslash$	$\backslash sqrt\{1\; -\; x^2\}$	$\{x\; \backslash over\; \backslash sqrt\{1\; +\; x^2\}\}$	$\{1\; \backslash over\; x\}$	$\{\backslash sqrt\{x^2\; -\; 1\}\; \backslash over\; x\}$	$\{1\; \backslash over\; \backslash sqrt\{1\; +\; x^2\}\}$
cos	$\backslash sqrt\{1\; -\; x^2\}$	$x\backslash$	$\{1\; \backslash over\; \backslash sqrt\{1\; +\; x^2\}\}$	$\{\backslash sqrt\{x^2\; -\; 1\}\; \backslash over\; x\}$	$\{1\; \backslash over\; x\}$	$\{x\; \backslash over\; \backslash sqrt\{1\; +\; x^2\}\}$
tan	$\{x\; \backslash over\; \backslash sqrt\{1\; -\; x^2\}\}$	$\{\backslash sqrt\{1\; -\; x^2\}\; \backslash over\; x\}$	$x\backslash$	$\{1\; \backslash over\; \backslash sqrt\{x^2\; -\; 1\}\}$	$\backslash sqrt\{x^2\; -\; 1\}$	$\{1\; \backslash over\; x\}$
csc	$\{1\; \backslash over\; x\}$	$\{1\; \backslash over\; \backslash sqrt\{1\; -\; x^2\}\}$	$\{\backslash sqrt\{1\; +\; x^2\}\; \backslash over\; x\}$	$x\backslash$	$\{x\; \backslash over\; \backslash sqrt\{x^2\; -\; 1\}\}$	$\backslash sqrt\{1\; +\; x^2\}$
sec	$\{1\; \backslash over\; \backslash sqrt\{1\; -\; x^2\}\}$	$\{1\; \backslash over\; x\}$	$\backslash sqrt\{1\; +\; x^2\}$	$\{x\; \backslash over\; \backslash sqrt\{x^2\; -\; 1\}\}$	$x\backslash$	$\{\backslash sqrt\{1\; +\; x^2\}\; \backslash over\; x\}$
cot	$\{\backslash sqrt\{1\; -\; x^2\}\; \backslash over\; x\}$	$\{x\; \backslash over\; \backslash sqrt\{1\; -\; x^2\}\}$	$\{1\; \backslash over\; x\}$	$\backslash sqrt\{x^2\; -\; 1\}$	$\{1\; \backslash over\; \backslash sqrt\{x^2\; -\; 1\}\}$	$x\backslash$

One procedure that can be used to obtain the elements of this table is as follows:
Given trigonometric functions φ and ψ, what is φ(arcψ(x)) equal to?

1. Find an equation that relates φ(u) and ψ(u) to each other:

   $f(\backslash varphi(u),\; \backslash psi(u))\; =\; 0\; \backslash$

2. Let u = arc ψ(x), so that:

   $f(\backslash varphi(\{\backslash rm\; arc\}\backslash psi(x)),\backslash psi(\{\backslash rm\; arc\}\backslash psi(x))\; =\; 0\; \backslash$

   $f(\backslash varphi(\{\backslash rm\; arc\}\backslash psi(x)),x)\; =\; 0\; \backslash$

3. Solve the last equation for φ(arcψ(x)).

Example. What is cot(arccsc(x)) equal to? First, find an equation which relations the functions cot and csc to each other, such as

$\backslash cot^2\; u\; +\; 1\; =\; \backslash csc^2\; u\; \backslash$.

Second, let u = arccsc(x):

$\backslash cot^2(\backslash arccsc(x))\; +\; 1\; =\; \backslash csc^2(\backslash arccsc(x))\; \backslash$,

$\backslash cot^2(\backslash arccsc(x))\; +\; 1\; =\; x^2\; \backslash$.

Third, solve this equation for cot(arccsc(x)):

$\backslash cot^2(\backslash arccsc(x))\; =\; x^2\; -\; 1,\; \backslash$

$\backslash cot(\backslash arccsc(x))\; =\; \backslash pm\backslash sqrt\{x^2\; -\; 1\},$

and this is the formula which shows up in the sixth row and fourth column of the table.

Exponential forms

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

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

where $i^\{2\}=-1.\backslash ,$

Infinite product formulae

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For applications to special functions, the following infinite product formulae for trigonometric functions are useful:

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

$\backslash sinh\; x\; =\; x\; \backslash prod\_\{n\; =\; 1\}^\backslash infty\backslash left(1\; +\; \backslash frac\{x^2\}\{\backslash pi^2\; n^2\}\backslash right)$

$\backslash cos\; x\; =\; \backslash prod\_\{n\; =\; 1\}^\backslash infty\backslash left(1\; -\; \backslash frac\{x^2\}\{\backslash pi^2(n\; -\; \backslash frac\{1\}\{2\})^2\}\backslash right)$

$\backslash cosh\; x\; =\; \backslash prod\_\{n\; =\; 1\}^\backslash infty\backslash left(1\; +\; \backslash frac\{x^2\}\{\backslash pi^2(n\; -\; \backslash frac\{1\}\{2\})^2\}\backslash right)$

$\backslash frac\{\backslash sin\; x\}\{x\}\; =\; \backslash prod\_\{n\; =\; 1\}^\backslash infty\backslash cos\backslash left(\backslash frac\{x\}\{2^n\}\backslash right)$

The Gudermannian function

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The Gudermannian function relates the circular and hyperbolic trigonometric functions without resorting to complex numbers; see that article for details.

Identities without variables

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Richard Feynman is reputed to have learned as a boy, and always remembered, the following curious identity:

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

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

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

The following is perhaps not as readily generalized to an identity containing variables:

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

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

$\backslash cos\backslash left(\; \backslash frac\{2\backslash pi\}\{21\}\backslash right)$

\,+\, \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 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.

An efficient way to compute π is based on the following identity without variables, due to Machin:

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

or, alternatively, by using Euler's formula:

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

Under this heading, there are also "special values" of trigonometric functions, including the ones that every student of trigonometry learns:

$$

\begin{matrix} \sin 0 & = & \sin 0^\circ & = & 0 & = & \cos 90^\circ & = & \cos \left( \frac {\pi} {2} \right) \\ \\ \sin \left( \frac {\pi} {6} \right) & = & \sin 30^\circ & = & 1/2 & = & \cos 60^\circ & = & \cos \left( \frac {\pi} {3} \right) \\ \\ \sin \left( \frac {\pi} {4} \right) & = & \sin 45^\circ & = & \sqrt{2}/2 & = & \cos 45^\circ & = & \cos \left( \frac {\pi} {4} \right) \\ \\ \sin \left( \frac {\pi} {3} \right) & = & \sin 60^\circ & = & \sqrt{3}/2 & = & \cos 30^\circ & = & \cos \left( \frac {\pi} {6} \right) \\ \\ \sin \left( \frac {\pi} {2} \right) & = & \sin 90^\circ & = & 1 & = & \cos 0^\circ & = & \cos 0 \end{matrix}

$\backslash sin\{\backslash frac\{\backslash pi\}\{7\}\}=\backslash frac\{\backslash sqrt\{7\}\}\{6\}-$

\frac{\sqrt{7}}{189} \sum_{j=0}^{\infty} \frac{(3j+1)!}{189^j j!\,(2j+2)!} \!

$\backslash sin\{\backslash frac\{\backslash pi\}\{18\}\}=$

\frac{1}{6} \sum_{j=0}^{\infty} \frac{(3j)!}{27^j j!\,(2j+1)!} \!

With the golden ratio φ:

$\backslash cos\; \backslash left(\; \backslash frac\; \{\backslash pi\}\; \{5\}\; \backslash right)\; =\; \backslash cos\; 36^\backslash circ=\{\backslash sqrt\{5\}+1\; \backslash over\; 4\}\; =\; \backslash varphi\; /2$

$\backslash sin\; \backslash left(\; \backslash frac\; \{\backslash pi\}\; \{10\}\; \backslash right)\; =\; \backslash sin\; 18^\backslash circ\; =\; \{\backslash sqrt\{5\}-1\; \backslash over\; 4\}\; =\; \{\backslash varphi\; -\; 1\; \backslash over\; 2\}\; =\; \{1\; \backslash over\; 2\backslash varphi\}$

Calculus

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In calculus the relations stated below require angles to be measured in radians; the relations would become more complicated if angles were measured in another unit such as degrees. If the trigonometric functions are defined in terms of geometry, then their derivatives can be found by verifying two limits. The first is:

$\backslash lim\_\{x\backslash rightarrow\; 0\}\backslash frac\{\backslash sin(x)\}\{x\}=1,$

verified using the unit circle and squeeze theorem. It may be tempting to propose to use L'Hôpital's rule to establish this limit. However, if one uses this limit in order to prove that the derivative of the sine is the cosine, and then uses the fact that the derivative of the sine is the cosine in applying L'Hôpital's rule, one is reasoning circularly—a logical fallacy. The second limit is:

$\backslash lim\_\{x\backslash rightarrow\; 0\}\backslash frac\{1-\backslash cos(x)\}\{x\}=0,$

verified using the identity tan(x/2) = (1 − cos(x))/sin(x). Having established these two limits, one can use the limit definition of the derivative and the addition theorems to show that sin′(x) = cos(x) and cos′(x) = −sin(x). If the sine and cosine functions are defined by their Taylor series, then the derivatives can be found by differentiating the power series term-by-term.

$\{d\; \backslash over\; dx\}\backslash sin(x)\; =\; \backslash cos(x)$

The rest of the trigonometric functions can be differentiated using the above identities and the rules of differentiation. We have:

$\{d\; \backslash over\; dx\}\backslash cos(x)\; =\; -\backslash sin(x)$

$\{d\; \backslash over\; dx\}\backslash tan(x)\; =\; \backslash sec^2(x)$

$\{d\; \backslash over\; dx\}\backslash cot(x)\; =\; -\backslash csc^2(x)$

$\{d\; \backslash over\; dx\}\backslash sec(x)\; =\; \backslash sec(x)\; \backslash tan(x)$

$\{d\; \backslash over\; dx\}\backslash csc(x)\; =\; -\; \backslash csc(x)\backslash cot(x)$

$\{d\; \backslash over\; dx\}\backslash arcsin(x)=\backslash frac\{1\}\{\backslash sqrt\{1-x^2\}\}$

$\{d\; \backslash over\; dx\}\backslash arctan(x)=\backslash frac\{1\}\{1+x^2\}$

The integral identities can be found in Wikipedia's table of integrals.

Implications

The fact that the differentiation of trigonometric functions (sine and cosine) results in linear combinations of the same two functions is of fundamental importance to many fields of mathematics, including differential equations and fourier transformations.

Geometric proofs

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sin(x + y) = sin(x) cos(y) + cos(x) sin(y)