7.\\
\includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-5_648_1590_296_275}

The circle $C _ { 1 }$ has centre $O$ and radius $R$. The tangents $A P$ and $B P$ to $C _ { 1 }$ meet at the point $P$ and angle $A P B = 2 \alpha , 0 < \alpha < \frac { \pi } { 2 }$. A sequence of circles $C _ { 1 } , C _ { 2 } , \ldots , C _ { n } , \ldots$ is drawn so that each new circle $C _ { n + 1 }$ touches each of $C _ { n } , A P$ and $B P$ for $n = 1,2,3 , \ldots$ as shown in Figure 2. The centre of each circle lies on the line $O P$.
\begin{enumerate}[label=(\alph*)]
\item Show that the radii of the circles form a geometric sequence with common ratio

$$\frac { 1 - \sin \alpha } { 1 + \sin \alpha }$$
\item Find, in terms of $R$ and $\alpha$, the total area enclosed by all the circles, simplifying your answer.

The area inside the quadrilateral $P A O B$, not enclosed by part of $C _ { 1 }$ or any of the other circles, is $S$.
\item Show that

$$S = R ^ { 2 } \left( \alpha + \cot \alpha - \frac { \pi } { 4 } \operatorname { cosec } \alpha - \frac { \pi } { 4 } \sin \alpha \right) .$$
\item Show that, as $\alpha$ varies,

$$\frac { \mathrm { d } S } { \mathrm {~d} \alpha } = R ^ { 2 } \cot ^ { 2 } \alpha \left( \frac { \pi } { 4 } \cos \alpha - 1 \right)$$
\item Find, in terms of $R$, the least value of $S$ for $\frac { \pi } { 6 } \leq \alpha \leq \frac { \pi } { 4 }$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2006 Q7 [20]}}