Sum of Reciprocals of Squares of Odd Integers/Proof 7

Proof
By Half-Range Fourier Cosine Series for Identity Function over $\openint 0 \pi$:


 * $\ds x = \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\map \cos {2 n - 1} x} {\paren {2 n - 1}^2}$

for $x \in \openint 0 \pi$.

We have that:
 * $\map f \pi = \map f {\pi - 2 \pi} = \map f {-\pi} = \pi$

and so:
 * $\map f {\pi^-} = \map f {\pi^+}$

Hence we can set $x = \pi$: