Difficult Induction Question Using Triginometry and Inequalities

please put me out of my misery and give some hints as to how to complete this one:

Given that

$$\sum_{k=1}^n x_k < \frac{\pi}{2}$$

Where $0\leq x_i \leq \frac{\pi}{2}$ for $i=1,2,3,4,\ldots,n$

Prove by mathematical induction that for $n=1,2,3,\ldots$ that:

$$\tan(x_1+\ldots+x_n)\geq \tan x_1 + \ldots + \tan x_n$$