Find the series for $\csc^2(x)$

Question

In a problem I'm being asked to find the series representation for $$\csc^2(x)=\sum_{-\infty}^{\infty} \frac{1}{(x-n \pi)^{2}}$$ I'm supposed to solve this problem by using complex analysis. However, I have no idea where to start.
How do I approach this problem?

I tried using the residue theorem, however, I didn't get very far.

Answer 1

The log-derivative of $\sin x=x\prod_{n\ge1}(1-x^2/(n\pi)^2)$ is$$\cot x=\frac1x+\sum_{n\ge1}\frac{2x}{x^2-(n\pi)^2}=\sum_{n\in\Bbb Z}\frac{1}{x-n\pi}$$by partial fractions. Now apply $-\frac{d}{dx}$.