# Trigonometric Identities

### Fundamental Formulae

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

### Double Angle Formulae

$\sin(2\theta)=2\sin(\theta)\cos(\theta)$
$\cos(2\theta)=\cos(\theta)^2-\sin^2(\theta)$
$\cos(2\theta)=2\cos^2(\theta)-1$
$\cos(2\theta)=1-2\sin^2(\theta)$

### Compound Angle Formulae

$\sin(\alpha\pm\beta)=\sin(\alpha)\cos(\beta)\pm\cos(\alpha)\sin(\beta)$
$\cos(\alpha\pm\beta)=\sin(\alpha)\sin(\beta)\mp\cos(\alpha)\cos(\beta)$
$\tan(\alpha\pm\beta)=\frac{\tan(\alpha)\pm\tan(\beta)}{1\mp\tan(\alpha)\tan(\beta)}$

Example 1:  Simplify

$\frac{\sqrt{1-\cos^2(x)}}{\cos(x)}$

Using the identity

$\sin^2(x)+\cos^2(x)=1$

we have

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

and so

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

Hence,

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

Example 2: Simplify

$\frac{\tan(\theta)}{\sec^2(\theta)-2}$

This can be rewritten as

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

Multiplying the top and bottom of the fraction by $\cos^2(x)$ gives

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

Using the double angle formulae, this becomes

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