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 $\triangle \mathrm{ABC}$, find the value of $\sum a(\sin B-\sin C)$.

Solution:

Using sine rule, we have

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$

$\Rightarrow a=k \sin A, b=k \sin B, c=k \sin C$

$\therefore \sum a(\sin B-\sin C)$

$=\sum k \sin A(\sin B-\sin C)$

$=k \sum \sin A(\sin B-\sin C)$

$=k[\sin A(\sin B-\sin C)+\sin B(\sin C-\sin A)+\sin C(\sin A-\sin B)]$

$=k(\sin A \sin B-\sin A \sin C+\sin B \sin C-\sin B \sin A+\sin C \sin A-\sin C \sin B)$

$=k \times 0=0$

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