Sum of Reciprocals of Squares of Odd Integers/Proof 7

Proof
By Fourier Cosine Series for $x$ over $\left[{0 \,.\,.\, \pi}\right]$:


 * $\displaystyle x = \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac{\cos \left({2 n - 1}\right) x} {\left({2 n - 1}\right)^2}$

for $x \in \left[{0 \,.\,.\, \pi}\right]$.

We have that:
 * $f \left({\pi}\right) = f \left({\pi - 2 \pi}\right) = f \left({-\pi}\right) = \pi$

and so:
 * $f \left({\pi^-}\right) = f \left({\pi^+}\right)$

Hence we can set $x = \pi$: