(i) Express \(\cos \theta + 2 \sin \theta\) in the form \(R \cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). Give the exact values of \(R\) and \(\tan \alpha\).

(ii) Hence, showing all necessary working, show that \(\int_0^{\frac{1}{4}\pi} \frac{15}{(\cos \theta + 2 \sin \theta)^2} \, d\theta = 5.\)