Answer each of the following questions in one word or one sentence or as per exact requirement of for question:

$In any triangle ABC, find the value of a s i n (B - C) + b s i n (C - A) + c s i n (A - B .)$

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Solution

$a s i n (B - C) + b s i n (C - A) + c s i n (A - B .) = k s i n A s i n (B - C) + k s i n B s i n (C - A) + k s i n C s i n (A - B) = k s i n (π - (B + C)) s i n (B - C) + k s i n (π - (C + A)) s i n (C - A) + k s i n (π - (A + B)) s i n (A - B) = k s i n (B + C) s i n (B - C) + k s i n (C + A) s i n (C - A) + k s i n (A + B) s i n (A - B) = k [s i n^{2} B - s i n^{2} C + s i n^{2} C - s i n^{2} A + s i n^{2} A - s i n^{2} B] = k \times 0 = 0$

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