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Question
Answer the following questions in one word or one sentence or as per exact requirement of the question.
In any ∆ABC, find the value of
\[\sum^{}_{}a\left( \text{ sin }B - \text{ sin }C \right)\]
Solution
Using sine rule, we have
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k\]
\[ \Rightarrow a = k\sin A, b = k\sin B, c = k\sin C\]
\[\therefore \sum^{}_{}a\left( \sin B - \sin C \right)\]
\[ = \sum^{}_{}k\sin A\left( \sin B - \sin C \right)\]
\[= k\sum^{}_{}\sin A\left( \sin B - \sin C \right)\]
\[= k\left[ \sin A\left( \sin B - \sin C \right) + \sin B\left( \sin C - \sin A \right) + \sin C\left( \sin A - \sin B \right) \right]\] \[= k\left( \sin A\sin B - \sin A\sin C + \sin B\sin C - \sin B\sin A + \sin C\sin A - \sin C\sin B \right)\]
\[= k\times0=0\]
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