Question

If $\cos \alpha +\cos \beta =0=\sin \alpha +\sin \beta$, then $\cos 2\alpha +\cos 2\beta =$

• A. $-2\sin (\alpha +\beta )$
• B. $-2\cos (\alpha +\beta )$
• C. $2\sin (\alpha +\beta )$
• D. $2\cos (\alpha +\beta )$

Answer 1

The correct option is B $-2\cos (\alpha +\beta )$

We have, $\cos \alpha +\cos \beta =0=\sin \alpha +\sin \beta$

Squaring both sides, we get

${\cos }^2\alpha +{\cos }^2\beta +2\cos \alpha \cos \beta ={\sin }^2\alpha +{\sin }^2\beta +2\sin \alpha \sin \beta$

$\Rightarrow ({\cos }^2\alpha -{\sin }^2\alpha )+({\cos }^2\beta -{\sin }^2\beta )$

$=-2(\cos \alpha \cos \beta -\sin \alpha \sin \beta )$

$\Rightarrow \cos 2\alpha +\cos 2\beta =-2\cos (\alpha +\beta )$