$a^{2} s i n (B - C) = (b^{2} - c^{2}) s i n A$

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Solution

$a^{2} s i n (B - C) = (b^{2} - c^{2}) s i n A = \frac{s i n A}{a} = \frac{s i n B}{b} = \frac{s i n C}{c} = k L H S a^{2} s i n (B - C) = a^{2} (s i n B . c o s C - s i n C . c o s B) = a^{2} k b . \frac{a^{2} + b^{2} - c^{2}}{2 a b} - a^{2} c k . \frac{a^{2} + c^{2} - b^{2}}{2 a c} [Using cos rule and sine rule] = a^{2} k . \frac{a^{2} + b^{2} - c^{2}}{2 a} - a^{2} k . \frac{a^{2} + c^{2} - b^{2}}{2 a} = a^{2} k . (\frac{a^{2} + b^{2} - c^{2} - a^{2} - c^{2} + b^{2}}{2 a}) = a^{2} k . (\frac{2 b^{2} - 2 c^{2}}{2 a}) = a k . (b^{2} - c^{2}) = s i n A (b^{2} - c^{2}) = R H S Hence proved.$

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