Question

$\cos 2\theta \cos 2\varphi +{\sin }^2\left(\theta -\varphi \right)-{\sin }^2\left(\theta +\varphi \right)=\cos \left(2\theta +2\varphi \right)$

Answer 1

${\sin }^{2}A-{\sin }^{2}B=\sin (A+B)\sin (A-B)\to \left(1\right)$

$\text{LHS}$

$\cos \left(2\theta \right)\cos 2\varphi +{\sin }^2(\theta -\varphi )-{\sin }^2(\theta +\varphi )$

$A=\theta -\varphi$

$B=\theta +\varphi$

$\Rightarrow \cos \left(2\theta \right)\cos 2\varphi +\sin \left(2\theta \right)\sin \left(2\varphi \right)$

$\Rightarrow \cos (2\theta -2\varphi )$ =$\text{RHS}$

Hence proved