7 (a) Show that the curve with Cartesian equation
\(\left(x^{2}+y^{2}\right)^{\frac{5}{2}}=4 x y\left(x^{2}-y^{2}\right)\)
has polar equation \(r=\sin 4 \theta\).

The curve \(C\) has polar equation \(r=\sin 4 \theta\), for \(0 \leqslant \theta \leqslant \frac{1}{4} \pi\).
(b) Sketch \(C\) and state the equation of the line of symmetry.

(c) Find the exact value of the area of the region enclosed by \(C\).

(d) Using the identity \(\sin 4 \theta \equiv 4 \sin \theta \cos ^{3} \theta-4 \sin ^{3} \theta \cos \theta\), find the maximum distance of \(C\) from the line \(\theta=\frac{1}{2} \pi\). Give your answer correct to 2 decimal places.