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

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The residue of the function $\cot^{n}(z)$ at the point $z=0$ is determined to be $\sin \left( \frac{n \pi}{2}\right)$ for natural numbers $n$. This result arises from analyzing the Laurent series expansion of $\cot(z)$ around $z=0$. The behavior of $\cot(z)$ near this point contributes to the calculation of the residue. The hint provided suggests focusing on the properties of sine and the periodic nature of the cotangent function. This establishes a clear relationship between the residue and the sine function evaluated at specific multiples of $\pi$.
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Show that the residue of $\cot^{n}(z)$ at $z=0$ is $\sin \left( \frac{n \pi}{2}\right)$, $n \in \mathbb{N}$.
 
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Hint:

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$.
 

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