(i)[3]
Express $5\cos \theta - 2\sin \theta$ as $R\cos(\theta + \alpha)$, with $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$. State the value of $\alpha$ correct to 4 decimal places.
(ii)[5]
Using your answer from part (i), solve $5\cot \theta - 4\cosec \theta = 2$ for $0 < \theta < 2\pi$.
(iii)[3]
Evaluate $\int \frac{1}{\left(5\cos \frac{1}{2}x - 2\sin \frac{1}{2}x\right)^2} \, dx$.