# Tangent of Sum

## Theorem

$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$

where $\tan$ is tangent.

### Corollary

$\map \tan {a - b} = \dfrac {\tan a - \tan b} {1 + \tan a \tan b}$

## Proof

 $\displaystyle \map \tan {a + b}$ $=$ $\displaystyle \frac {\map \sin {a + b} } {\map \cos {a + b} }$ Tangent is Sine divided by Cosine $\displaystyle$ $=$ $\displaystyle \frac {\sin a \cos b + \cos a \sin b} {\cos a \cos b - \sin a \sin b}$ Sine of Sum and Cosine of Sum $\displaystyle$ $=$ $\displaystyle \frac {\frac {\sin a} {\cos a} + \frac {\sin b} {\cos b} } {1 - \frac {\sin a \sin b} {\cos a \cos b} }$ dividing top and bottom by $\cos a \cos b$ $\displaystyle$ $=$ $\displaystyle \frac {\tan a + \tan b} {1 - \tan a \tan b}$ Tangent is Sine divided by Cosine

$\blacksquare$