For which of the following functions the $\sum_{x\in S(f)}\frac{1}{x}$ converges? For real valued function $f$ define 
$$S(f)=\{x:x>0,f(x)=x\}$$
For which of the following functions the $\sum_{x\in S(f)}\frac{1}{x}$ converges?
$\tan x,\tan^2x,\tan{\sqrt{x}},\sqrt{\tan x},\tan 2x$
I do not have any idea to solve this problem,I think all these have infinitely many fixed points.I need help.
Thanks.

Hint: for $f(x)=\tan x$,
$\tan x=x$ has at least one solution (which we call $x_k$) in each interval of the form $(2k\pi-\pi/2,2k\pi+\pi/2)$. So 
$$\sum_{x\in S(f)} \frac1x \geq \sum_{x\in S(f)} \frac{1}{x_k} \geq \sum_{k} \frac{1}{2k\pi+\pi/2}...$$
Similarly try for the other functions.
