Using Contour Integrals to Calculate Other Integrals How would I show that $$\int_{0}^{\pi /2}\sin^{2n}(t)\, dt = \frac{\pi}{2} \frac{1(3)(5)...(2n-1)}{2(4)(6)(8)...(2n)}$$
using contour integration? My guess is that I would integrate along the unit circle $|z| = 1$ but I'm not sure how to get any further.

First note that by symmetry your integral is $1/4$ the integral from $0$ to $2 \pi$.  Next, with $z = e^{it}$, $\sin(t) = (z - 1/z)/(2i)$ while $dt = dz/(iz)$.
Now find the residue...
