4. (a) Given that \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), express \(16 \cos ^ { 4 } \theta\) in the form \(a \cos 4 \theta + b \cos 2 \theta + c\), where \(a , b , c\) are integers whose values are to be determined. [5]
The diagram below shows a sketch of the curve C with polar equation $$r = 3 - 4 \cos ^ { 2 } \theta , \quad \text { where } \frac { \pi } { 6 } \leqslant \theta \leqslant \frac { 5 \pi } { 6 }$$ Initial line
(b) Calculate the area of the region enclosed by the curve \(C\).
(c) Find the exact polar coordinates of the points on \(C\) at which the tangent is perpendicular to the initial line.
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