The even function $f(x)$ has the Fourier cosine series

$$f(x)=\frac{1}{2} a_{0}+\sum_{n=1}^{\infty} a_{n} \cos n x$$

in the interval $-\pi \leqslant x \leqslant \pi$. Show that

$$\frac{1}{\pi} \int_{-\pi}^{\pi}(f(x))^{2} d x=\frac{1}{2} a_{0}^{2}+\sum_{n=1}^{\infty} a_{n}^{2}$$

Find the Fourier cosine series of $x^{2}$ in the same interval, and show that

$$\sum_{n=1}^{\infty} \frac{1}{n^{4}}=\frac{\pi^{4}}{90}$$