Solve \(\frac{\cos5\theta}{\cos\theta}=0\) for \(0\le\theta<\pi\), giving the answers as \(\frac{n\pi}{10}\) with \(n\in\mathbb N\).[5]
The identities \(\cos3\theta=4\cos^3\theta-3\cos\theta\) and \(\sin3\theta=3\sin\theta-4\sin^3\theta\) are given. Use \(\cos(A+B)\) to prove \(\frac{\cos5\theta}{\cos\theta}=16\sin^4\theta-12\sin^2\theta+1\).[7]
Find the four exact values of \(\sin\theta\) satisfying \(16\sin^4\theta-12\sin^2\theta+1=0\).[4]
Using the preceding results, show that \(\sin\frac{\pi}{10}=\sqrt{\frac{3-\sqrt5}{8}}\).[2]
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