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Trigonometric Identities

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

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

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

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

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 :

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

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

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