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Show that cos2 A + cos2 B – 2 cos A cos B cos(A + B) = sin2(A + B). |
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Answer» LHS = cos2 A + cos2 B – 2 cos A cos B [cos A cos B – sin A sin B] = cos2 A + cos2 B – 2 cos2 A cos2 B + 2 sin A cos A sin B cos B = (cos2 A – cos2 A cos2 B) + (cos2 B – cos2 A cos2 B) + 2 sin A cos A sin B cos B = cos2 A (1 – cos2 B) + cos2 B (1 – cos2 A) + 2 sin A cos A sin B cos B = cos2 A sin2 B + cos2 B sin2 A + 2 sin A cos B sin B cos A = (sin A cos B + cos A sin B)2 = sin2 (A + B) = RHS |
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