(i)[2]
Show that $\frac{\cos 2\theta}{1 + \cos 2\theta}$ can be written as $1 - \frac{1}{2} \sec^2 \theta$.
(ii)[4]
Solve the equation $\frac{\cos 2\alpha}{1 + \cos 2\alpha} = 13 + 5 \tan \alpha$ within $0 < \alpha < \pi$.
(iii)[4]
Find the exact value for $\int_{-\frac{1}{2}\pi}^{\frac{1}{2}\pi} \frac{\cos x}{1 + \cos x} \, dx$.