7. $\left[ \arccos x \right.$ and $\arctan x$ are alternative notation for $\cos ^ { - 1 } x$ and $\tan ^ { - 1 } x$ respectively $]$

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{fc5d0d07-b750-4646-bdcb-419a290200c9-5_387_935_322_566}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}

Figure 2 shows a sketch of the curve $C _ { 1 }$ with equation $y = \cos ( \cos x ) , 0 \leqslant x < 2 \pi$ .\\
The curve has turning points at $( 0 , \cos 1 ) , P , Q$ and $R$ as shown in Figure 2.
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the points $P , Q$ and $R$ .

The curve $C _ { 2 }$ has equation $y = \sin ( \cos x ) , 0 \leqslant x < 2 \pi$ .The curves $C _ { 1 }$ and $C _ { 2 }$ intersect at the points $S$ and $T$ .
\item Copy Figure 2 and on this diagram sketch $C _ { 2 }$ stating the coordinates of the minimum point on $C _ { 2 }$ and the points where $C _ { 2 }$ meets or crosses the coordinate axes.

The coordinates of $S$ are $( \alpha , d )$ where $0 < \alpha < \pi$ .
\item Show that $\alpha = \arccos \left( \frac { \pi } { 4 } \right)$ .
\item Find the value of $d$ in surd form and write down the coordinates of $T$ .

The tangent to $C _ { 1 }$ at the point $S$ has gradient $\tan \beta$ .
\item Show that $\beta = \arctan \sqrt { } \left( \frac { 16 - \pi ^ { 2 } } { 32 } \right)$ .
\item Find,in terms of $\beta$ ,the obtuse angle between the tangent to $C _ { 1 }$ at $S$ and the tangent to $C _ { 2 }$ at $S$ .
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2012 Q7 [24]}}