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Question
Prove that cos(A + B) cos(A – B) = cos2A – sin2B = cos2B – sin2A
Solution
L.H.S = cos(A + B) cos(A – B)
= (cos A cos B – sin A sin B)(cos A cos B + sin (A sin B)
= cos2A cos2B – sin2A sin2B
= cos2A (1 – sin2B) – (1 – cos2A) sin2B
= cos2A – cos2A sin2B – sin2B + cos2A sin2B
= cos2A – sin2B
= R.H.S
Now cos2A – sin2B = (1 – sin2A) – (1 – cos2B)
= 1 – sin2A – 1 + cos2B
= cos2B – sin2A
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