Use mean value theorem to prove :
$\displaystyle \mid tan \ x + tan \ y \mid \geq \mid x+y \mid \ , \forall x,y \in (\frac {-\pi}{2}, \frac {\pi}{2}) $
Suppose withou loss of generality that $\displaystyle u,v \in (-\pi/2,\pi.2)$ and $\displaystyle v>u$ then the MVT tells you that there is a $\displaystyle c$ in $\displaystyle (u,v)$ such that:
$\displaystyle \left. \frac{d}{d\theta}\tan(\theta)\right|_{\theta=c}=\f rac{\tan(u)-\tan(v)}{u-v}$
But
$\displaystyle \frac{d}{d\theta}\tan(\theta) \ge 1 \ \ \ \theta \in (-\pi/2,\pi/2)$
and you should using the fact that $\displaystyle \tan$ is odd be able to finish from there.
CB