4．A rectangle \(A B C D\) is drawn so that \(A\) and \(B\) lie on the \(x\)－axis，and \(C\) and \(D\) lie on the curve with equation \(y = \cos x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\) ．The point \(A\) has coordinates \(( p , 0 )\) ，where \(0 < p < \frac { \pi } { 2 }\) ．
（a）Find an expression，in terms of \(p\) ，for the area of this rectangle． The maximum area of \(A B C D\) is \(S\) and occurs when \(p = \alpha\) ．Show that
（b）\(\frac { \pi } { 4 } < \alpha < 1\) ，
（c）\(S = \frac { 2 \alpha ^ { 2 } } { \sqrt { } \left( 1 + \alpha ^ { 2 } \right) }\) ，
（d）\(\frac { \pi ^ { 2 } } { 2 \sqrt { } \left( 16 + \pi ^ { 2 } \right) } < S < \sqrt { } 2\) ．