I was studying the sinc function, it led me to the study the following equation on $\mathbb{R}$ $$ x=\tan\left(x\right) \ \ \ \ \left(\star\right) $$ The equation $\left(\star\right)$ has a unique solution $x_n$ on $\displaystyle I_n=\left]\frac{\pi}{2}+n\pi, \frac{\pi}{2}+\left(n+1\right)\pi\right[$ for $n \in \mathbb{Z}$ with $x_{n}=-x_{-n}$ and $x_0=0$, which allows us to define a sequence on $\mathbb{N}$ only.

I've read somewhere the following ( astonishing ) equality

$$ \sum_{n=0}^{+\infty}\frac{1}{\left(x_n\right)^2}=\frac{1}{10} $$

This sum does exist, because I've shown that $$ \frac{1}{\left(x_n\right)^2} \underset{(+\infty)}{\sim}\frac{1}{\pi^2 n^2} $$ But I dont know how to compute it. I thought about residue theorem making $x_n$ appears in a pôle but cannot find a way to prove it. Any help would be great.