Boundary Values & Fourier Series · Superchallenge problem

On $[-\pi,\pi]$, $\cosh x = \dfrac{\sinh\pi}{\pi} + \dfrac{2\sinh\pi}{\pi}\sum_{n\ge1}\dfrac{(-1)^n}{1+n^2}\cos nx$. Use this to find $\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{(1+n^2)^2}$ in closed form, and give its value as a decimal rounded to $4$ places.