A) \[<1\]
B) \[=-1\]
C) \[<0\]
D) None of these
Correct Answer: A
Solution :
We have,\[sin\text{ }\alpha +sin\,\,\beta +sin\text{ }\gamma -sin\text{ }(\alpha +\beta +\gamma )\] \[=sin\text{ }\alpha +sin\text{ }\beta +sin\text{ }\gamma -sin\alpha \text{ }cos\beta \text{ }cos\gamma \] \[-cos\text{ }\alpha \text{ }sin\text{ }\beta \text{ }cos\text{ }\gamma -cos\text{ }\alpha \text{ }cos\text{ }\beta \text{ }sin\text{ }\gamma \] \[+\text{ }sin\text{ }\alpha \text{ }sin\text{ }\beta \text{ }sin\text{ }\gamma \] \[=\sin \alpha (1-\cos \beta \cos \gamma )+\sin \beta (1-\cos \alpha \cos \gamma )\] \[+\sin \gamma (1-\cos \alpha \cos \beta )+\sin \alpha \sin \beta \sin \gamma \] \[\therefore \] \[\sin \alpha +\sin \beta +\sin \gamma >\sin (\alpha +\beta +\gamma )\] \[\Rightarrow \] \[\frac{\sin (\alpha +\beta +\gamma )}{\sin \alpha +\sin \beta +\sin \gamma }<1\]
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