(i)[3]
Differentiate $\frac{1}{\cos \theta}$ to show that, when $y = \sec \theta$, then $\frac{dy}{d\theta} = \tan \theta \sec \theta$.
(ii)[4]
Hence demonstrate that $\frac{d^2y}{d\theta^2} = a\sec^3 \theta + b\sec \theta$, and state the values of $a$ and $b$.
(iii)[5]
Find the exact value for $\int_0^{\frac{\pi}{4}} (1 + \tan^2 \theta - 3\sec \theta \tan \theta)\,d\theta$.