6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dfb57dc0-5831-4bbb-b1e5-58c4798215cb-5_700_684_246_694}
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\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of the curve \(C\) with parametric equations
$$x = 2 \sin t , \quad y = \ln ( \sec t ) , \quad 0 \leqslant t < \frac { \pi } { 2 }$$
The tangent to \(C\) at the point \(P\) ，where \(t = \frac { \pi } { 3 }\) ，cuts the \(x\)－axis at \(A\) ．
（a）Show that the \(x\)－coordinate of \(A\) is \(\frac { \sqrt { } 3 } { 3 } ( 3 - \ln 2 )\) ．
The shaded region \(R\) lies between \(C\) ，the positive \(x\)－axis and the tangent \(A P\) as shown in Figure 2 ．
（b）Show that the area of \(R\) is \(\sqrt { 3 } ( 1 + \ln 2 ) - 2 \ln ( 2 + \sqrt { 3 } ) - \frac { \sqrt { 3 } } { 6 } ( \ln 2 ) ^ { 2 }\) ．