# Difference of Arctangents

## Theorem

$\arctan a - \arctan b = \arctan \dfrac {a - b} {1 + a b}$

where $\arctan$ denotes the arctangent.

## Proof

Let $x = \arctan a$ and $y = \arctan b$.

Then:

 $(1):\quad$ $\displaystyle \tan x$ $=$ $\displaystyle a$ $(2):\quad$ $\displaystyle \tan y$ $=$ $\displaystyle b$ $\displaystyle \map \tan {\arctan a - \arctan b}$ $=$ $\displaystyle \map \tan {x - y}$ $\displaystyle$ $=$ $\displaystyle \frac {\tan x - \tan y} {1 + \tan x \tan y}$ Tangent of Difference $\displaystyle$ $=$ $\displaystyle \frac {a - b} {1 + a b}$ by $(1)$ and $(2)$ $\displaystyle \leadsto \ \$ $\displaystyle \arctan a - \arctan b$ $=$ $\displaystyle \arctan \frac {a - b} {1 + a b}$

$\blacksquare$