11. The infinite series C and S are defined by
$$\begin{aligned}
& \mathrm { C } = \cos \theta + \frac { 1 } { 2 } \cos 5 \theta + \frac { 1 } { 4 } \cos 9 \theta + \frac { 1 } { 8 } \cos 13 \theta + \ldots
& \mathrm { S } = \sin \theta + \frac { 1 } { 2 } \sin 5 \theta + \frac { 1 } { 4 } \sin 9 \theta + \frac { 1 } { 8 } \sin 13 \theta + \ldots
\end{aligned}$$
Given that the series C and S are both convergent,
- show that
$$C + i S = \frac { 2 e ^ { i \theta } } { 2 - e ^ { 4 i \theta } }$$
- Hence show that
$$\mathrm { S } = \frac { 4 \sin \theta + 2 \sin 3 \theta } { 5 - 4 \cos 4 \theta }$$
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