Prove by mathematical induction that, for \(z \neq 1\) and all positive integers \(n\),
$$1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 } = \frac { z ^ { n } - 1 } { z - 1 }$$
By letting \(z = \frac { 1 } { 2 } ( \cos \theta + \mathrm { i } \sin \theta )\), use de Moivre's theorem to deduce that
$$\sum _ { m = 1 } ^ { \infty } \left( \frac { 1 } { 2 } \right) ^ { m } \sin m \theta = \frac { 2 \sin \theta } { 5 - 4 \cos \theta }$$