(a)[4]
Prove that $\cos(\theta + 30^\circ)\cos(\theta + 60^\circ) = \frac{1}{4}\sqrt{3} - \frac{1}{2}\sin 2\theta$.
(b)[4]
Solve the equation $5\cos(2\alpha + 30^\circ)\cos(2\alpha + 60^\circ) = 1$ for $0^\circ < \alpha < 90^\circ$.
(c)[3]
Show that the exact value of $\cos 20^\circ \cos 50^\circ + \cos 40^\circ \cos 70^\circ$ is $\frac{1}{2}\sqrt{3}$.