(i)[4]
Starting from the expansion of $\cos(2x + x)$, show that $\cos 3x = 4\cos^3 x - 3\cos x$.
(ii)[2]
Hence determine the solutions to the equation $\cos 3x + 3\cos x + 1 = 0$, for $0 \leq x \leq \pi$.
(iii)[4]
Determine the exact value of $\int_{\pi/6}^{\pi/3} \cos^3 x\, dx$.