A curve \(C\) has polar equation \(r^{2}=8 \operatorname{cosec} 2 \theta\) for \(0\lt \theta\lt \frac{1}{2} \pi\). Find a cartesian equation of \(C\).

Sketch \(C\).

Determine the exact area of the sector bounded by the arc of \(C\) between \(\theta=\frac{1}{6} \pi\) and \(\theta=\frac{1}{3} \pi\), the half-line \(\theta=\frac{1}{6} \pi\) and the half-line \(\theta=\frac{1}{3} \pi\).
[It is given that \(\int \operatorname{cosec} x \mathrm{~d} x=\ln \left|\tan \frac{1}{2} x\right|+c\).]