Evaluate: sin (beta + gamma - alpha ) + sin ( | vedclass

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(B) Let the expression be $E = \sin (\beta + \gamma - \alpha ) + \sin (\gamma + \alpha - \beta ) + \sin (\alpha + \beta - \gamma ) - \sin (\alpha + \b

Vedclass · Accepted Answer

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

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(B) Let the expression be $E = \sin (\beta + \gamma - \alpha ) + \sin (\gamma + \alpha - \beta ) + \sin (\alpha + \beta - \gamma ) - \sin (\alpha + \beta + \gamma )$.Using the sum-to-product formula $\sin C + \sin D = 2 \sin \frac{C+D}{2} \cos \frac{C-D}{2}$:Group the first two terms: $\sin (\beta + \gamma - \alpha ) + \sin (\gamma + \alpha - \beta ) = 2 \sin \gamma \cos (\beta - \alpha )$.Group the last two terms: $\sin (\alpha + \beta - \gamma ) - \sin (\alpha + \beta + \gamma ) = 2 \cos (\alpha + \beta ) \sin (-\gamma) = -2 \sin \gamma \cos (\alpha + \beta )$.Combining these,$E = 2 \sin \gamma [\cos (\beta - \alpha ) - \cos (\alpha + \beta )]$.Using the identity $\cos (A-B) - \cos (A+B) = 2 \sin A \sin B$:$E = 2 \sin \gamma [2 \sin \alpha \sin \beta] = 4 \sin \alpha \sin \beta \sin \gamma $.

Evaluate: $\sin (\beta + \gamma - \alpha ) + \sin (\gamma + \alpha - \beta ) + \sin (\alpha + \beta - \gamma ) - \sin (\alpha + \beta + \gamma )$

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