8 Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that
$$1 + z = 2 \cos \frac { 1 } { 2 } \theta \left( \cos \frac { 1 } { 2 } \theta + \mathrm { i } \sin \frac { 1 } { 2 } \theta \right)$$
By considering \(( 1 + z ) ^ { n }\), where \(n\) is a positive integer, deduce the sum of the series
$$\binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \ldots + \binom { n } { n } \sin n \theta$$