- (a) Write down and simplify the Maclaurin series for \(\sin 2 x\) as far as the term in \(x ^ { 5 }\).
(b) Using your answer to part (a), determine the Maclaurin series for \(\cos ^ { 2 } x\) as far as the term in \(x ^ { 4 }\). - (a) Show that \(\tan \theta\) may be expressed as \(\frac { 2 t } { 1 - t ^ { 2 } }\), where \(t = \tan \left( \frac { \theta } { 2 } \right)\).
The diagram below shows a sketch of the curve \(C\) with polar equation
$$r = \cos \left( \frac { \theta } { 2 } \right) , \quad \text { where } - \pi < \theta \leqslant \pi .$$
\includegraphics[max width=\textwidth, alt={}, center]{1d01b3d3-8a45-4c64-9d2f-ced042d8fba3-4_680_887_1171_580}
(b) Show that the \(\theta\)-coordinate of the points at which the tangent to \(C\) is perpendicular to the initial line satisfies the equation
$$\tan \theta = - \frac { 1 } { 2 } \tan \left( \frac { \theta } { 2 } \right) .$$
(c) Hence, find the polar coordinates of the points on \(C\) where the tangent is perpendicular to the initial line.
(d) Calculate the area of the region enclosed by the curve \(C\) and the initial line for \(0 \leqslant \theta \leqslant \pi\).