Prove by induction that, if \(n\) is a positive integer,
$$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$
Hence, given that
$$z = \cos \theta + \mathrm { i } \sin \theta$$
show that
$$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$
Given further that \(z + \frac { 1 } { z } = \sqrt { 2 }\), find the value of
$$z ^ { 10 } + \frac { 1 } { z ^ { 10 } }$$