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MCQ

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

A
$\sin \,\,2\alpha $
B
$\cos \,\,2\beta $
✓
$\cos \,\,2\alpha $
D
$\sin \,\,2\beta $

✓

Answer

Correct option: C.

$\cos \,\,2\alpha $

c
(c) $\cos 2(\alpha + \beta ) = 2{\cos ^2}(\alpha + \beta ) - 1,$

$2{\sin ^2}\beta = 1 - \cos 2\beta $

$L.H.S.$ $ = - \cos 2\beta + 2\cos (\alpha + \beta )\,[2\sin \alpha \sin \beta + \cos (\alpha + \beta )]$

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

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

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