9 In this question you must show detailed reasoning.
- Show that \(\mathrm { e } ^ { \mathrm { i } \theta } - \mathrm { e } ^ { - \mathrm { i } \theta } = 2 \mathrm { i } \sin \theta\).
- Hence, show that \(\frac { 2 } { \mathrm { e } ^ { 2 \mathrm { i } \theta } - 1 } = - ( 1 + \mathrm { i } \cot \theta )\).
- Two series, \(C\) and \(S\), are defined as follows.
$$\begin{aligned}
& C = 2 + 2 \cos \frac { \pi } { 10 } + 2 \cos \frac { \pi } { 5 } + 2 \cos \frac { 3 \pi } { 10 } + 2 \cos \frac { 2 \pi } { 5 }
& S = 2 \sin \frac { \pi } { 10 } + 2 \sin \frac { \pi } { 5 } + 2 \sin \frac { 3 \pi } { 10 } + 2 \sin \frac { 2 \pi } { 5 }
\end{aligned}$$
By considering \(C + \mathrm { i } S\), find a simplified expression for \(C\) in terms of only integers and \(\cot \frac { \pi } { 20 }\). - Verify that \(S = C - 2\) and, by considering the series in their original form, explain why this is so.
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