If α+β+γ=π , then the expression sin^2 α+sin^2 β-sin^2 γ has the value

Correct answer: (D) 2 sin α sin β cos γ. sin^2 α+sin^2 β-sin^2 γ =sin^2 α+sin (β+γ) sin (β-γ) =sin^2 α+sin (π-α) sin (β-γ) =sin α[sin α+sin (β-γ)] =sin α[sin (β+γ)+sin (β-γ)] =2 sin α sin β cos γ

Q57 / 150 · MHT CET 2024 (02 May Shift 2) · Mathematics

If $\alpha+\beta+\gamma=\pi$, then the expression $\sin ^2 \alpha+\sin ^2 \beta-\sin ^2 \gamma$ has the value

A
$2 \sin \alpha \sin \beta \sin \gamma$
B
$2 \cos \alpha \sin \beta \sin \gamma$
C
$2 \sin \alpha \cos \beta \sin \gamma$
Correct answer
D
$2 \sin \alpha \sin \beta \cos \gamma$

Solution

$\begin{aligned} & \sin ^2 \alpha+\sin ^2 \beta-\sin ^2 \gamma \\ & =\sin ^2 \alpha+\sin (\beta+\gamma) \sin (\beta-\gamma) \\ & =\sin ^2 \alpha+\sin (\pi-\alpha) \sin (\beta-\gamma) \\ & =\sin \alpha[\sin \alpha+\sin (\beta-\gamma)] \\ & =\sin \alpha[\sin (\beta+\gamma)+\sin (\beta-\gamma)] \\ & =2 \sin \alpha \sin \beta \cos \gamma\end{aligned}$

Hence, the correct answer is option: 'D'.

If $\alpha+\beta+\gamma=\pi$, then the expression $\sin ^2 \alpha+\sin ^2 \beta-\sin ^2 \gamma$ has the value

Solution

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