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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 triangle ABC, find the value of \[a\sin\left( B - C \right) + b\sin\left( C - A \right) + c\sin\left( A - B \right)\
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Solution
Using sine rule, we have
\[a\sin\left( B - C \right) + b\sin\left( C - A \right) + c\sin\left( A - B \right)\]
\[ = k\sin A\sin\left( B - C \right) + k\sin B\sin\left( C - A \right) + k\sin C\sin\left( A - B \right)\]
\[ = k\sin\left[ \pi - \left( B + C \right) \right]\sin\left( B - C \right) + k\sin\left[ \pi - \left( C + A \right) \right]\sin\left( C - A \right) + k\sin\left[ \pi - \left( A + B \right) \right]\sin\left( A - B \right)\]
\[= k\left[ \sin\left( B + C \right)\sin\left( B - C \right) + \sin\left( C + A \right)\sin\left( C - A \right) + \sin\left( A + B \right)\sin\left( A - B \right) \right]\]
\[ = k\left( \sin^2 B - \sin^2 C + \sin^2 C - \sin^2 A + \sin^2 A - \sin^2 B \right)\]
\[ = k \times 0\]
\[ = 0\]
Hence, the required value is 0.
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