( + - ) + ( + - ) + ( + - ) - ( + + ) = - Vidyadip

MCQ

$\sin (\beta + \gamma - \alpha ) + \sin (\gamma + \alpha - \beta )$$ + \sin (\alpha + \beta - \gamma ) - \sin (\alpha + \beta + \gamma ) = $

A
$2\sin \alpha \sin \beta \sin \gamma $
✓
$4\sin \alpha \sin \beta \sin \gamma $
C
$\sin \alpha \sin \beta \sin \gamma $
D
None of these

✓

Answer

Correct option: B.

$4\sin \alpha \sin \beta \sin \gamma $

b
(b) Combine first two terms and last two terms

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

$ = 2\,\sin \,\gamma \,[\cos \,(\beta - \alpha ) - \cos \,(\alpha + \beta )]$

$ = 2\,\sin \,\gamma \,.\,2\,\sin \alpha \,\sin \beta $

$ = 4\sin \alpha \sin \beta \sin \gamma $.

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