(a(i))[3]
Differentiate $\frac{\cos x}{\sin x}$ to demonstrate that, when $y = \cot x$, $\frac{dy}{dx} = -\csc^2 x$.
(a(ii))[4]
By writing $\cot^2 x$ in terms of $\csc^2 x$ and using the result from part (i), show that $\int_{\pi/4}^{3\pi/4} \cot^2 x\,dx = 1 - \tfrac{1}{4}\pi$.
(a(iii))[3]
Write $\cos 2x$ in terms of $\sin^2 x$ and hence show that $\frac{1}{1 - \cos 2x}$ may be written as $\tfrac{1}{2}\csc^2 x$. Hence, using the result from part (ii), evaluate $\int \frac{1}{1 - \cos 2x}\,dx$.