Riemann Zeta Function of 4/Proof 1

Proof
By Fourier Series of Fourth Power of x, for $x \in \closedint {-\pi} \pi$:
 * $\displaystyle x^4 = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \, \map \cos {n \pi} \, \map \cos {n x}$

Setting $x = \pi$: