$b s i n B - c s i n C = a s i n (B - C)$

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Solution

$\frac{s i n A}{a} = \frac{s i n B}{b} = \frac{s i n C}{c} = k R H S, a s i n (B - C) = a s i n B . c o s C - a s i n C . c o s B = a (b k) . (\frac{a^{2} + b^{2} - c^{2}}{2 a b}) - a (c k) . (\frac{a^{2} + c^{2} - b^{2}}{2 a c}) = k . \frac{(a^{2} + b^{2} - c^{2})}{2} - k \frac{(a^{2} + c^{2} - b^{2})}{2} = 2 k . \frac{(b^{2} - c^{2})}{2} = b . (k b) - c (k c) = b (s i n B) - c (s i n C) L H S .$

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