(i)[3]
Prove that $\tan \theta + \cot \theta$ equals $\frac{2}{\sin 2\theta}$.
(ii(a))[2]
Hence find the exact value for $\tan \frac{\pi}{8} + \cot \frac{\pi}{8}$.
(ii(b))[3]
Evaluate $\int_0^{\frac{\pi}{2}} \frac{6}{\tan \theta + \cot \theta} \, d\theta$.