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

• This may not be the proof by induction you are looking for, but the statement follows from convexity of $\tan x$ and Jensen's inequality. – mlc May 1 '17 at 14:10
• @mlc which is a far better proof if you ask me :) Post it as an answer so I can upvote it! Just title it appropriately to note you're not doing an inductive proof – Brevan Ellefsen May 1 '17 at 14:25

## 1 Answer

To prove in general that $f(\sum x_i) \geq \sum f(x_i)$, it is enough by induction to prove the case $n=2$. Now $\tan(x+y)=\frac{\sin x \cos y+\sin y \cos x}{\cos x \cos y -\sin x \sin y}\geq \frac{\sin x \cos y+\sin y \cos x}{\cos x \cos y}=\tan x + \tan y$ and we are done.