Question

If $\sin \theta$ is G.M of $\sin \varphi$ and $\cos \varphi$, then prove that $\cos 2\theta =2{\cos }^2(\frac{\pi }{4}+\varphi )$

Answer 1

${\sin }^2\theta =\sin \varphi \cos \varphi$
$2{\sin }^2\theta =\sin 2\varphi$
$1-\cos 2\theta =-\cos (\pi /2+2\varphi )$
$\therefore \cos 2\theta =1+\cos (\pi /2+2\varphi )$
$\cos 2\theta =2{\cos }^2(\frac{\pi }{4}+\varphi )$