Finding the simple poles and getting the sum of residues

[HACR]

Homework Statement

$$C_{N}$$ denotes the positively oriented boundary of the square whose edges lie along the lines $$x=+/- (N+1/2\pi), y=+/- (N+1/2\pi)$$.

Then show that [tex]\int_{C_{N}} \frac{dz}{z^{2}sin(z))=2\pi i[\frac{1}{6}+2\sum_{n=1}^{N}\frac{(-1)^n}{n^2{\pi}^2}]

Homework Equations

The Attempt at a Solution

So the problem boils down to finding the simple poles and getting the sum of residues. the first one occurs at z=0. But for this one, this is a double pole for z, and infinitely many zeros for sin(z) at infinity (pole at zero). So then how to determine the residue of the first pole?

 

[Asim Riaz]

I know that the residue of a double pole is the coefficient of (z-a)^-2 in the Laurent series expansion. But I'm not sure how to determine this coefficient.