Question 9 - A-Level Maths

By first expanding $\cos ( 2 \theta + \theta )$, prove that $$\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta$$
Hence prove that $$\cos 6 \theta \equiv 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
Show that the only solutions of the equation $$1 + \cos 6 \theta = 18 \cos ^ { 2 } \theta$$ are odd multiples of $90 ^ { \circ }$.