Trigonometric identity

A list of useful trigonometric identities. Generally used for manipulating equations.

Fundamental identities

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

Even-odd identities

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

Sine and cosine law

$$\frac{a}{\sin (A)}=\frac{b}{\sin (B)}=\frac{c}{\sin (C)}$$ $$\frac{\sin (A)}{a}=\frac{\sin (B)}{b}=\frac{\sin (C)}{c}$$ $$c^2=a^2+b^2-2ab\cos (C)$$ $$\cos (C)=\frac{a^2+b^2-c^2}{1ab}$$

Pythagorean identities

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

Double angle identities

We can derive the $\sin$ and $\cos$ identities with a technique involving the binomial theorem.

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

Compound angle identities

$$\sin (A\pm B)=\sin (A)\cos (B)\pm \cos (A)\sin (B)$$ $$\cos (A\pm B)=\cos (A)\cos (B)\mp \sin (A)\sin (B)$$ $$\tan (A\pm B)=\frac{\tan (A)\pm \tan (B)}{1\mp \tan (A)\tan (B)}$$