1. ## mean value theorem

Use mean value theorem to prove :
$\mid tan \ x + tan \ y \mid \geq \mid x+y \mid \ , \forall x,y \in (\frac {-\pi}{2}, \frac {\pi}{2})$

2. Originally Posted by flower3
Use mean value theorem to prove :
$\mid tan \ x + tan \ y \mid \geq \mid x+y \mid \ , \forall x,y \in (\frac {-\pi}{2}, \frac {\pi}{2})$
Suppose withou loss of generality that $u,v \in (-\pi/2,\pi.2)$ and $v>u$ then the MVT tells you that there is a $c$ in $(u,v)$ such that:

$\left. \frac{d}{d\theta}\tan(\theta)\right|_{\theta=c}=\f rac{\tan(u)-\tan(v)}{u-v}$

But

$\frac{d}{d\theta}\tan(\theta) \ge 1 \ \ \ \theta \in (-\pi/2,\pi/2)$

and you should using the fact that $\tan$ is odd be able to finish from there.

CB