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Question
Show that cos2 A + cos2 B – 2 cos A cos B cos(A + B) = sin2(A + B)
Solution
L.H.S = 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
= cos2A(1 – cos2 B) + cos2B(1 – cos2 A) + 2 sin A cos A sin B cos B
= cos2A sin2B + cos2B sin2A + 2 sin A cos B sin B cos A
= (sin A cos B + cos A sin B)2
= sin2 (A + B)
= R.H.S
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