(a) Using the residue theorem, evaluate

$$\int_{|z|=1}\left(z-\frac{1}{z}\right)^{2 n} \frac{d z}{z}$$

(b) Deduce that

$$\int_{0}^{2 \pi} \sin ^{2 n} t d t=\frac{\pi}{2^{2 n-1}} \frac{(2 n) !}{(n !)^{2}}$$