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$.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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