TITLE: Why is $\arctan\frac{x+y}{1-xy} = \arctan x +\arctan y$?
QUESTION [5 upvotes]: Why is $\arctan\frac{x+y}{1-xy} = \arctan x +\arctan y$?
It is said that this is derived from trigonometry, but I couldn't find why this is the case.

REPLY [3 votes]: Computing with complex numbers:
$$
(1+xi)(1+yi)=(1-xy)+(x+y)i
$$
Take arg on both sides
$$
\arctan x + \arctan y
= \arg(1+xi)+\arg(1+yi)
= \arg((1+xi)(1+yi))
\\
= \arg((1-xy)+(x+y)i)
= \arctan\frac{x+y}{1-xy}
$$