Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
Answer each of 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 $a \sin (B-C)+b \sin (C-A)+c \sin (A-B)$
Using sine rule, we have
$a \sin (B-C)+b \sin (C-A)+c \sin (A-B)$
$=k \sin A \sin (B-C)+k \sin B \sin (C-A)+k \sin C \sin (A-B)$
$=k \sin [\pi-(B+C)] \sin (B-C)+k \sin [\pi-(C+A)] \sin (C-A)+k \sin [\pi-(A+B)] \sin (A-B)$
$=k[\sin (B+C) \sin (B-C)+\sin (C+A) \sin (C-A)+\sin (A+B) \sin (A-B)]$
$=k\left(\sin ^{2} B-\sin ^{2} C+\sin ^{2} C-\sin ^{2} A+\sin ^{2} A-\sin ^{2} B\right)$
$=k \times 0$
$=0$
Hence, the required value is 0.
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