2013 JJC P2 Q1

A curve \(C\) has parametric equations

\(x=\left( 1-\sin t \right)\sin t,\quad y=\left( 1-\sin t \right)\cos t,\) for \(-\frac{\pi }{2}\le t\le \frac{\pi }{2}\).

1. Sketch \(C\), giving the coordinates of any points of intersection with the \(x\)- and \(y\)-axes.[3]
2. Show that the area of the region enclosed by \(C\) and the \(x\)-axis, \(A\) is \(\displaystyle\displaystyle \displaystyle\int_{\,-\alpha }^{\,\,\beta \,}{\left( \cos t-\sin t\cos t \right)\left( \cos t-2\sin t\cos t \right)\mathrm{d}t}+\displaystyle\displaystyle \displaystyle\int_{\,\,\alpha }^{\,\,\beta }{\left( \sin t-{{\sin }^{2}}t \right)\left( {{\sin }^{2}}t-{{\cos }^{2}}t-\sin t \right)}\ \mathrm{d}t,\) for some \(0\le \alpha ,\ \beta \le \frac{\pi }{2}\). Find the numerical value of \(A\), giving your answer correct to \(3\) decimal places.[4]