• # question_answer If $\alpha ,\,\beta ,\,\gamma \in \,\left( 0,\,\frac{\pi }{2} \right)$, then $\frac{\sin \,(\alpha +\beta +\gamma )}{\sin \alpha +\sin \beta +\sin \gamma }$ is   A) < 1 B) >1 C) = 1 D) None of these

We have $\sin \alpha +\sin \beta +\sin \gamma -\sin (\alpha +\beta +\gamma )$ $=\sin \alpha +\sin \beta +\sin \gamma -\sin \alpha \cos \beta \cos \gamma$ $-\cos \alpha \sin \beta \cos \gamma -\cos \alpha \cos \beta \sin \gamma +\sin \alpha \sin \beta \sin \gamma$ $=\sin \alpha (1-\cos \beta \cos \gamma )+\sin \beta (1-\cos \alpha \cos \gamma )$ $+\sin \gamma (1-\cos \alpha \cos \beta )+\sin \alpha \sin \beta \sin \gamma >0$ $\therefore \sin \alpha +\sin \beta +\sin \gamma >\sin (\alpha +\beta +\gamma )$ $\Rightarrow \frac{\sin (\alpha +\beta +\gamma )}{\sin \alpha +\sin \beta +\sin \gamma }<1$ .