Fourier Series

Usual exercises in Fourier series ask students to prove identity of the form \[
f=\sum a_n\cos nx+\sum b_n\sin nx.
\] Most of them are extremely straightforward and what you need is sufficient manpower, patience and time. If I were the one who teach Fourier series, I would propose the following identities as examples:

Problem. By using the Fourier series expansion of $f(x)=\cos \alpha x$, $x\in (-\pi,\pi)$, prove that \[
\cot x=\frac{1}{x}+\sum_{n=1}^\infty \brac{\frac{1}{x-n\pi}+\frac{1}{x+n\pi}}=\frac{1}{x}+\sum_{n=1}^\infty \frac{2x}{x^2-n^2\pi^2}
\] and also \[
\csc x=\frac{1}{x}+\sum_{n=1}^\infty (-1)^n \brac{\frac{1}{x-n\pi} +\frac{1}{x+n\pi}}=\frac{1}{x}+\sum_{n=1}^\infty (-1)^n \frac{2x}{x^2-n^2\pi^2}.
\] Also evaluate $\dis \sum_{n=1}^\infty \frac{1}{n^2-\alpha^2}$.

The surprise should be: there is apparently no evidence that Fourier series is involved!