Prove by induction that, if \(n\) is a positive integer,
$$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$
Find the value of \(\left( \cos \frac { \pi } { 6 } + \mathrm { i } \sin \frac { \pi } { 6 } \right) ^ { 6 }\).
Show that
$$( \cos \theta + \mathrm { i } \sin \theta ) ( 1 + \cos \theta - \mathrm { i } \sin \theta ) = 1 + \cos \theta + \mathrm { i } \sin \theta$$