(i)[2]
Prove that when $y = \frac{1}{\cos \theta}$, then $\frac{dy}{d\theta} = \sec \theta \tan \theta$.
(ii)[3]
Prove that $\frac{1 + \sin \theta}{1 - \sin \theta} = 2\sec^2 \theta + 2\sec \theta \tan \theta - 1$.
(iii)[4]
Hence determine the exact value of $\int_{0}^{\frac{\pi}{4}} \frac{1 + \sin \theta}{1 - \sin \theta} \, d\theta$.
(c(iii))[4]
Hence determine the exact value of $\displaystyle \int_{0}^{\frac{\pi}{4}} \frac{1 + \sin \theta}{1 - \sin \theta} \, d\theta$.