(a) The curve \(C\) has polar equation \(r^2=\tan 2\theta\), where \(0\le\theta\le\frac18\pi\). Sketch \(C\) and state the greatest distance of a point on \(C\) from the pole.

(b) Find the exact value of the area of the region bounded by \(C\) and the half-line \(\theta=\frac18\pi\).

(c) Show that \(C\) has Cartesian equation \(x^4-2xy-y^4=0\), given the first-quadrant restrictions from \(0\le\theta\le\frac18\pi\).

(d) Using your answer to part (b), deduce the exact value of the area bounded by \(C\), the \(x\)-axis and the line \(x=\cos\frac18\pi\).