(i)[3]
Write $\cos\theta + 2\sin\theta$ in the form $R\cos(\theta - \alpha)$, where $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$. State the exact values of $R$ and $\tan\alpha$.
(ii)[5]
Hence, with all working shown, demonstrate that $\int_{0}^{4\pi} \frac{15}{(\cos\theta + 2\sin\theta)^2}\, d\theta = 5$.