Answer each of the following questions in one word or one sentence or as per exact requirement of the question.
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 $a \sin (B-C)+b \sin (C-A)+c \sin (A-B)$

Solution:

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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