List of trigonometric identities

A trigonometric identity is an equation involving trigonometric functions which is 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: With trigonometric functions, we define functions sin², cos², etc., such that sin²(x) = (sin(x))².

From the Definitions

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

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

Periodicity, Symmetry and Shifts

These are easiest proven from the unit circle:

$\sin(x)=\sin(x+2\pi )\qquad \cos(x)=\cos(x+2\pi )\qquad \tan(x)=\tan(x+\pi )$

$\sin(-x)=-\sin(x)\qquad \cos(-x)=\cos(x)$

$\tan(-x)=-\tan(x)\qquad \cot(-x)=-\cot(x)$

$\sin(x)=\cos \left({\frac {\pi }{2}}-x\right)\qquad \cos(x)=\sin \left({\frac {\pi }{2}}-x\right)\qquad \tan(x)=\cot \left({\frac {\pi }{2}}-x\right)$

From the Pythagorean Theorem

$\sin ^{2}(x)+\cos ^{2}(x)=1\qquad \tan ^{2}(x)+1=\sec ^{2}(x)\qquad \cot ^{2}(x)+1=\csc ^{2}(x)$

Addition Theorems

The quickest way to prove these is Euler's formula. The tangent formula follows from the other two.

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

$\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$

$\tan(x+y)={\frac {\tan(x)+\tan(y)}{1-\tan(x)\tan(y)}}$

Double-Angle Formulas

These can be shown by substituting $x=y$ in the addition theorems, and using the Pythagorean formula for the latter two. Or use de Moivres formula with $n=2$.

$\sin(2x)=2\sin(x)\cos(x)$

$\cos(2x)=\cos ^{2}(x)-\sin ^{2}(x)=2\cos ^{2}(x)-1=1-2\sin ^{2}(x)$

$\tan(2x)={\frac {2\tan(x)}{(1-\tan ^{2}(x))}}$

$\cos(nx)+i\sin(nx)=(\cos(x)+i\sin(x))^{n}$

Multiple-Angle Formulas

If T_n is the nth Chebyshev polynomial then

$\cos(nx)=T_{n}(\cos(x)).$

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

Power-Reduction Formulas

Solve the third and fourth double angle formula for cos²(x) and sin²(x).

$\cos ^{2}(x)={1+\cos(2x) \over 2}$

$\sin ^{2}(x)={1-\cos(2x) \over 2}$

Half-Angle Formulas

Substitute x/2 for x in the power reduction formulas, then solve for cos(x/2) and sin(x/2).

$\left|\cos \left({\frac {x}{2}}\right)\right|={\sqrt {\left({\frac {1+\cos(x)}{2}}\right)}}$

$\left|\sin \left({\frac {x}{2}}\right)\right|={\sqrt {\left({\frac {1-\cos(x)}{2}}\right)}}$

Multiply tan(x/2) by 2cos(x/2) / ( 2cos(x/2)) and substitute sin(x/2) / cos(x/2) for tan(x/2). The numerator is then sin(x) via the double angle formula, and the denominator is 2cos²(x/2) - 1 + 1 which is cos(x) + 1 by the double angle formulae. The second formula comes from the first formula multiplied by sin(x) / sin(x) and simplified using the pythagorean trig identity.

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

Products to Sums

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

$\cos(x)\cos(y)={\cos(x+y)+\cos(x-y) \over 2}$

$\sin(x)\sin(y)={\cos(x-y)-\cos(x+y) \over 2}$

$\sin(x)\cos(y)={\sin(x+y)+\sin(x-y) \over 2}$

Sums to Products

Replace x by (x + y) / 2 and y by (x – y) / 2 in the Product-to-Sum formulas.

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

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

Calculus

If the trigonometric functions are defined in terms of geometry, then their derivatives can be found by first verifying that

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

and

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

and then using the limit definition of the derivative and the addition theorems; if they are defined by their Taylor series, then the derivatives can be found by differentiating the power series term by term.

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

The rest of the trig functions can be differentiated using the above identities and the rules of differentiation, for instance

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

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

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