(a) Show that \(\sum_{r=1}^{n} z^{4 r}=\frac{z^{4 n+2}-z^{2}}{z^{2}-z^{-2}}\), for \(z^{2} \neq z^{-2}\).
(b) By letting \(z=\cos \theta+\mathrm{i} \sin \theta\), show that, if \(\sin 2 \theta \neq 0\),
\(\sum_{r=1}^{n} \sin (4 r \theta)=\frac{\cos 2 \theta-\cos (4 n+2) \theta}{2 \sin 2 \theta} .\)