Question

Prove that $\cos 2\alpha =2{\sin }^2\beta +4\cos (\alpha +\beta )\sin \alpha \sin \beta +\cos 2(\alpha +\beta )$

Answer 1

$2{\sin }^2\beta +4\cos (\alpha +\beta )\sin \alpha \sin \beta +\cos 2(\alpha +\beta )=2{\sin }^2\beta +2\cos (\alpha +\beta )\left(\cos (\alpha -\beta )-\cos (\alpha +\beta )\right)+\cos 2(\alpha +\beta )=2{\sin }^2\beta +2\cos (\alpha +\beta )\cos (\alpha -\beta )-2{\cos }^2(\alpha +\beta )+\cos 2(\alpha +\beta )=2{\sin }^2\beta +2{\cos }^2\alpha -2{\sin }^2\beta -2{\cos }^2(\alpha +\beta )-2{\cos }^2(\alpha +\beta )-1=2{\cos }^2\alpha -1=\cos 2\alpha$