list of trigonometric identities

a trig identity is an equation that involves trig functions

remembering trig identities can help you simplify some expressions

this page is just a list

for visualization of these identities, see unit circle
for a better explanation of these identities, see here

reciprocal and quotient identities

from the definitions of the trig functions we can express all trig functions with just sine and cosine

$\tan (a)=\frac{\sin (a)}{\cos (a)}$

$\csc (a)=\frac{1}{\sin (a)}$

$\sec (a)=\frac{1}{\cos (a)}$

$\cot (a)=\frac{\cos (a)}{\sin (a)}$

Pythagorean identities

Pythagorean identities can be found by applying the Pythagorean theorem to the stuff in the unit circle

${\sin }^2(a)+{\cos }^2(a)=1$

${\tan }^2(a)+1={\sec }^2(a)$

${\cot }^2(a)+1={\csc }^2(a)$

Pythagorean identities are able to be rearranged to suit many simplifying purposes

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

${\cos }^2(a)=\frac{1+\cos (2a)}{2}$

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

angle sum and difference identities

$\sin (a+b)=\sin (a)\cdot \cos (b)+\cos (a)\cdot \sin (b)$

$\sin (a-b)=\sin (a)\cdot \cos (b)-\cos (a)\cdot \sin (b)$

$\cos (a+b)=\cos (a)\cdot \cos (b)-\sin (a)\cdot \sin (b)$

$\cos (a-b)=\cos (a)\cdot \cos (b)+\sin (a)\cdot \sin (b)$

$\tan (a+b)=\frac{\tan (a)+\tan (b)}{1-\tan (a)\tan (b)}$
$\tan (a-b)=\frac{\tan (a)-\tan (b)}{1+\tan (a)\tan (b)}$

double angle identities

these are derived from the angle sum identities and Pythagorean identities

$\sin (2a)=2\cdot \sin (a)\cdot \cos (a)$

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

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

half angle identities

these are derived from the double angle identities

$\sin (\frac{a}{2})=\pm \sqrt{\frac{1-\cos (a)}{2}}$

$\cos (\frac{a}{2})=\pm \sqrt{\frac{1+\cos (a)}{2}}$

$\tan (\frac{a}{2})=\frac{\sin (a)}{1+\cos (a)}$

symmetry and periodicity identities

$\sin (-a)=-\sin (a)$
$\cos (-a)=\cos (a)$
$\tan (-a)=-\tan (a)$

$\sin (a+2\pi )=\sin (a)$
$\cos (a+2\pi )=\cos (a)$
$\tan (a+\pi )=\tan (a)$

other identities

for some reason i have these

$\tan (\theta +\frac{\pi }{2})=\frac{1}{\tan (\theta )}$