If alpha, beta, gamma in left(0, frac{pi}{2}r | vedclass

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(A) Given $\alpha, \beta, \gamma \in \left(0, \frac{\pi}{2}\right)$.We know that for $x \in \left(0, \frac{\pi}{2}\right)$,$\sin x > 0$ and $\cos x <

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(A) Given $\alpha, \beta, \gamma \in \left(0, \frac{\pi}{2}\right)$.We know that for $x \in \left(0, \frac{\pi}{2}\right)$,$\sin x > 0$ and $\cos x Consider the expression $E = \sin \alpha + \sin \beta + \sin \gamma - \sin(\alpha + \beta + \gamma)$.Using the expansion $\sin(\alpha + \beta + \gamma) = \sin \alpha \cos \beta \cos \gamma + \cos \alpha \sin \beta \cos \gamma + \cos \alpha \cos \beta \sin \gamma - \sin \alpha \sin \beta \sin \gamma$,we get:$E = \sin \alpha(1 - \cos \beta \cos \gamma) + \sin \beta(1 - \cos \alpha \cos \gamma) + \sin \gamma(1 - \cos \alpha \cos \beta) + \sin \alpha \sin \beta \sin \gamma$.Since $\alpha, \beta, \gamma \in \left(0, \frac{\pi}{2}\right)$,we have $\cos \alpha, \cos \beta, \cos \gamma  0$,$(1 - \cos \alpha \cos \gamma) > 0$,and $(1 - \cos \alpha \cos \beta) > 0$.Thus,$E > 0$,which implies $\sin \alpha + \sin \beta + \sin \gamma > \sin(\alpha + \beta + \gamma)$.Therefore,$\frac{\sin(\alpha + \beta + \gamma)}{\sin \alpha + \sin \beta + \sin \gamma} < 1$.

If $\alpha, \beta, \gamma \in \left(0, \frac{\pi}{2}\right)$,then $\frac{\sin(\alpha + \beta + \gamma)}{\sin \alpha + \sin \beta + \sin \gamma}$ is

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