(i)[4]
Prove that $\cos 4\theta + 4\cos 2\theta = 8\cos^4 \theta - 3$.
(ii(a))[3]
Hence solve the equation $\cos 4\theta + 4\cos 2\theta = 1$ for $-\dfrac{1}{2}\pi \leq \theta \leq \dfrac{1}{2}\pi$.
(ii(b))[3]
Find the exact value for $\int_0^{\tfrac{1}{4}\pi} \cos^4 \theta\, d\theta$.