4. The curve \(C\) has parametric equations
$$\begin{gathered}
x = \cos ^ { 2 } t
y = \cos t \sin t
\end{gathered}$$
where \(0 \leqslant t < \pi\)
- Show that \(C\) is a circle and find its centre and its radius.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12d3f92f-8464-4ba1-93a2-c7b841e3d3de-3_668_750_726_660}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of \(C\). The point \(P\), with coordinates \(\left( \cos ^ { 2 } \alpha , \cos \alpha \sin \alpha \right) , \quad 0 < \alpha < \frac { \pi } { 2 }\), lies on \(C\). The rectangle \(R\) has one side on the \(x\)-axis, one side on the \(y\)-axis and \(O P\) as a diagonal, where \(O\) is the origin. - Show that the area of \(R\) is \(\sin \alpha \cos ^ { 3 } \alpha\)
- Find the maximum area of \(R\), as \(\alpha\) varies.