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If $\alpha ,\,\beta ,\,\gamma \in \,\left( {0,\,\frac{\pi }{2}} \right)$, then $\frac{{\sin \,(\alpha + \beta + \gamma )}}{{\sin \alpha + \sin \beta + \sin \gamma }}$ is
$< 1$
$> 1$
$= 1$
None of these
Solution
(a) We have $\sin \alpha + \sin \beta + \sin \gamma – \sin (\alpha + \beta + \gamma )$
$ = \sin \alpha + \sin \beta + \sin \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 $
$ = \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 > 0$
$\therefore \sin \alpha + \sin \beta + \sin \gamma > \sin (\alpha + \beta + \gamma )$
$ \Rightarrow \frac{{\sin (\alpha + \beta + \gamma )}}{{\sin \alpha + \sin \beta + \sin \gamma }} < 1$ .
