2) If A+B+C =180° prove that sin? A+sin? B + sin? C = 2(1+cos A cos B cos C)
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Let S = sin2A + sin2B + sin2C
so that 2S = 2sin2A + 1 – cos2B +1 – cos2C = 2 sin2A + 2 – 2cos(B + C) cos(B – C) = 2 – 2 cos2A + 2 – 2cos(B + C) cos(B – C) ∴ S = 2 + cosA [cos(B – C) + cos(B+ C)] since cosA = – cos(B+C)
∴ S = 2 + 2 cos A cos B cos C
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