Đặt $\arctan x = u, \arctan y = v$

$\Rightarrow x = \tan u, y = \tan v$

Xét $\tan(u+v) = \frac{tanu +tanv}{1-tanu.tanv}$

$\Leftrightarrow \tan(u+v) = \frac{x+y}{1-xy}$

$\Leftrightarrow u+v = \arctan \frac{x+y}{1-xy}$

$\Leftrightarrow \arctan x + \arctan y = \arctan \frac{x+y}{1-xy}$ (đpcm)