Question

$\cos 5\theta =16{\cos }^5\theta -20{\cos }^3\theta +5\cos \theta$

Answer 1

We know that,

$\cos 5\theta =16{\cos }^5\theta -20{\cos }^3\theta +5\cos \theta$

Here,

$\cos 5\theta =\cos \left(3\theta +2\theta \right)=\cos 3\theta .\sin 2\theta -\sin 3\theta .\sin 2\theta =\cos 3\theta .\cos 2\theta -\frac{1}{2}\left[\cos \theta -\cos 5\theta \right]=\left(4{\cos }^3\theta -3\cos \theta \right)\left(2{\cos }^2\theta -1\right)-\frac{1}{2}\cos \theta +\frac{1}{2}\cos 5\theta$

Now,

$\frac{\cos 5\theta }{2}=8{\cos }^5\theta -6{\cos }^3\theta -4{\cos }^3\theta +3\cos \theta -\frac{\cos \theta }{2}\frac{\cos 5\theta }{2}=8{\cos }^5\theta -10{\cos }^5\theta +5\cos \theta \cos 5\theta =16{\cos }^5\theta -20{\cos }^3\theta +5\cos \theta$

Hence, proved.