A curve \(C\) has parametric equations \(x=\theta-\sin\theta\), \(y=1-\cos\theta\), where \(0\le\theta\le2\pi\).
Show that \(\frac{\mathrm dy}{\mathrm dx}=\cot\frac12\theta\) and find the gradient of \(C\) at the point where \(\theta=\pi\). What can be said about the tangents to \(C\) as \(\theta\to0\) and \(\theta\to2\pi\)?[5]
Sketch \(C\), showing clearly the features of the curve at the points where \(\theta=0\), \(\pi\) and \(2\pi\).[3]
Without using a calculator, find the exact area of the region bounded by \(C\) and the \(x\)-axis.[5]
A point \(P\) on \(C\) has parameter \(p\), where \(0<p<\frac12\pi\). Show that the normal to \(C\) at \(P\) crosses the \(x\)-axis at the point with coordinates \((p,0)\).[3]