MHB Residue Calc: $\cot^n(z)$ at $z=0$

[polygamma]

Show that the residue of $\cot^{n}(z)$ at $z=0$ is $\sin \left( \frac{n \pi}{2}\right)$, $n \in \mathbb{N}$.

 

[polygamma]

Hint:

Spoiler

Integrate $\cot^{n} (z)$ around a rectangular contour with vertices at $- \frac{\pi}{2} - iR$, $ \frac{\pi}{2} - i R$, $\frac{\pi}{2} + iR$, and $- \frac{\pi}{2} + iR$. Then let $ R \to \infty$.