Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that \(z - \frac { 1 } { z } = 2 \mathrm { i } \sin \theta\) and hence express \(16 \sin ^ { 5 } \theta\) in the form \(\sin 5 \theta + p \sin 3 \theta + q \sin \theta\), where \(p\) and \(q\) are integers to be determined.