Quotient of sin 5 theta by sin theta as a quadratic in cos 2 theta, giving the exact value of cos 2 pi over 5

Solve the equation \(\frac{\sin5\theta}{\sin\theta}=0\) for \(0<\theta<\pi\), giving your answers in the form \(\frac{k\pi}{5}\), where \(k\) is a positive integer.[5]

You may use the identities \(\sin3\theta=3\sin\theta-4\sin^3\theta\) and \(\cos3\theta=4\cos^3\theta-3\cos\theta\).

By writing \(5\theta\) as \(3\theta+2\theta\), prove that \(\frac{\sin5\theta}{\sin\theta}=4\cos^22\theta+2\cos2\theta-1\).[7]

Find the two exact values of \(\cos2\theta\) that satisfy the equation \(4\cos^22\theta+2\cos2\theta-1=0\).[4]

Using your answers to parts (a) and (c), show that \(\cos\frac{2\pi}{5}=\frac{\sqrt5-1}{4}\).[2]