In Ito, McKean: Diffusion processes and their sample paths there is a problem(page 22, problem 3), where they they define Brownian motion as follows:
$$x(t):=\frac{t}{\sqrt{\pi}}g_{0}+ \sum_{n\geq 0} \sum_{k=2^{n-1}}^{2^n-1} \sqrt\frac{2}{\pi}\frac{\sin(kt)}{k}g_k $$
$g_k$-s are iid with $\mathcal{N}(0,1)$.
They prove continuity in full details, but after that they mention(without a proof) one also needs:
$$ \mathbb{E}(x(t)x(s))=\frac{ts}{\pi}+\frac{2}{\pi}\sum_{k\geq 1}\frac{\sin(kt)\sin(ks)}{k^2}=s \wedge{t}$$
Now, I understood the first equality, but somehow I fail to see the second. I tried to work with Parseval's identity, but did not achieve anything.
Any help appreciated, to understand this equality.
Thanks in advance!