Question

If $\alpha+\beta+\gamma=\pi$, then the value of $\sin ^{2} \alpha+\sin ^{2} \beta-\sin ^{2} \gamma$ is equal to

• A. $ 2\sin \alpha $
• B. $ 2\sin \alpha \cos\beta \sin\gamma $
• C. $ 2\sin \alpha \sin \beta \cos \gamma $
• D. $ 2\sin \alpha \sin \beta \sin \gamma $

Answer 1

Answer: (C)

$\therefore \sin ^{2} \alpha+\sin (\beta-\gamma) \sin (\beta+\gamma)$
$=\sin ^{2} \alpha+\sin (\pi-\alpha) \sin (\beta-\gamma)$
$=\sin \alpha[\sin \alpha+\sin (\beta-\gamma)]$
$=\sin \alpha[\sin (\pi-(\beta+\gamma))+\sin (\beta-\gamma)]$
$=\sin \alpha[\sin (\beta-\gamma)+\sin (\beta+\gamma)]$
$=2 \sin \alpha \sin \beta \cos \gamma$