# Nice proofs of ζ(4)=π490\zeta(4) = \frac{\pi^4}{90}?

I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman’s list or the answers to the question “Different methods to compute $\sum_{n=1}^{\infty} \frac{1}{n^2}$?”

Are there any nice ways to prove that

I already know some proofs that give all values of $\zeta(n)$ for positive even integers $n$ (like #7 on Robin Chapman’s list or Qiaochu Yuan’s answer in the linked question). I’m not so much interested in those kinds of proofs as I am those that are specifically for $\zeta(4)$.

I would be particularly interested in a proof that isn’t an adaption of one that $\zeta(2) = \pi^2/6$.

In the same spirit of the 1st proof of this answer. If we substitute $\pi$ for $x$ in the Fourier trigonometric series expansion of $% f(x)=x^{4}$, with $-\pi \leq x\leq \pi$,