8 (i) Prove by mathematical induction that, for $z \neq 1$ and all positive integers $n$,

$$1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 } = \frac { z ^ { n } - 1 } { z - 1 }$$

(ii) By letting $z = \frac { 1 } { 2 } ( \cos \theta + \mathrm { i } \sin \theta )$, use de Moivre's theorem to deduce that

$$\sum _ { m = 1 } ^ { \infty } \left( \frac { 1 } { 2 } \right) ^ { m } \sin m \theta = \frac { 2 \sin \theta } { 5 - 4 \cos \theta }$$

\hfill \mbox{\textit{CAIE FP1 2019 Q8}}