Trigonometric Identities

Trigonometric identities are equations involving the trigonometric functions that are true for every value of the variables involved.

Some of the most commonly used trigonometric identities are derived from the Pythagorean Theorem , like the following:

${\sin }^2\left(x\right)+{\cos }^2\left(x\right)=1$

$1+{\tan }^2\left(x\right)={\sec }^2\left(x\right)$

$1+{\cot }^2\left(x\right)={\csc }^2\left(x\right)$

There are also the reciprocal identities :

$\sin \left(x\right)=\frac{1}{\csc \left(x\right)}$       $\cos \left(x\right)=\frac{1}{\sec \left(x\right)}$       $\tan \left(x\right)=\frac{1}{\cot \left(x\right)}$

$\csc \left(x\right)=\frac{1}{\sin \left(x\right)}$       $\sec \left(x\right)=\frac{1}{\cos \left(x\right)}$       $\cot \left(x\right)=\frac{1}{\tan x}$

The quotient identities :

$\tan \left(u\right)=\frac{\sin \left(u\right)}{\cos \left(u\right)}$

$\cot \left(u\right)=\frac{\cos \left(u\right)}{\sin \left(u\right)}$

The co-function identities :

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

$\csc \left(\frac{\pi }{2}-x\right)=\sec \left(x\right)$       $\sec \left(\frac{\pi }{2}-x\right)=\csc \left(x\right)$       $\cot \left(\frac{\pi }{2}-x\right)=\tan \left(x\right)$

The even-odd identities :

$\sin \left(-x\right)=-\sin \left(x\right)$       $\cos \left(-x\right)=\cos \left(x\right)$       $\tan \left(-x\right)=-\tan \left(x\right)$

$\csc \left(-x\right)=-\csc \left(x\right)$       $\sec \left(-x\right)=\sec \left(x\right)$       $\cot \left(-x\right)=-\cot \left(x\right)$

The Bhaskaracharya sum and difference formulas :

$\sin \left(u\pm v\right)=\sin \left(u\right)\cos \left(v\right)+\cos \left(u\right)\sin \left(v\right)$

$\cos \left(u\pm v\right)=\cos \left(u\right)\cos \left(v\right)\mp \sin \left(u\right)\sin \left(v\right)$

$\tan \left(u\pm v\right)=\frac{\tan \left(u\right) \pm \tan \left(v\right)}{1 \mp \tan \left(u\right)\tan \left(v\right)}$

The double-angle formulas :

(These are really just special cases of Bhaskaracharya's formulas, when $u=v$ .)

$\sin \left(2u\right)=2\sin u\cos u$

$\cos \left(2u\right)={\cos }^2\left(u\right)-{\sin }^2\left(u\right)$

$=2{\cos }^2\left(u\right)-1$

$=1-{\sin }^2\left(u\right)$

$\tan \left(2u\right)=\frac{2\tan \left(u\right)}{1 - {\tan }^2\left(u\right)}$

The half-angle or power-reducing formulas :

(Again, a special case of Bhaskaracharya.)

${\sin }^2\left(u\right)=\frac{1 - \cos \left(2u\right)}{2}$

${\cos }^2\left(u\right)=\frac{1 + \cos \left(2u\right)}{2}$

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

The sum-to-product formulas :

$\sin \left(u\right)+\sin \left(v\right)=2\sin \left(\frac{u+v}{2}\right)\cos \left(\frac{u-v}{2}\right)$

$\sin \left(u\right)-\sin \left(v\right)=2\cos \left(\frac{u+v}{2}\right)\sin \left(\frac{u-v}{2}\right)$

$\cos \left(u\right)+\cos \left(v\right)=2\cos \left(\frac{u+v}{2}\right)\cos \left(\frac{u-v}{2}\right)$

$\cos \left(u\right)-\cos \left(v\right)=-2\sin \left(\frac{u+v}{2}\right)\sin \left(\frac{u-v}{2}\right)$

And the product-to-sum formulas :

$\sin \left(u\right)\sin \left(v\right)=\frac{1}{2}\left[\cos \left(u-v\right)-\cos \left(u+v\right)\right]$

$\cos \left(u\right)\cos \left(v\right)=\frac{1}{2}\left[\cos \left(u-v\right)+\cos \left(u+v\right)\right]$

$\sin \left(u\right)\cos \left(v\right)=\frac{1}{2}\left[\sin \left(u+v\right)+\sin \left(u-v\right)\right]$