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Question
Prove the following:
(cos x + cos y)2 + (sin x – sin y)2 = `4cos^2 ((x + y))/2`
Solution
L.H.S. = (cos x + cos y)2 + (sin x – sin y)2
= cos2 x + 2 cos x cos y + cos2 y + sin2x – 2 sin x sin y + sin2 y
= (cos2 x + sin2 x) + (cos2 y + sin2 y) + 2(cos x cos y – sin x sin y)
= 1 + 1 + 2 cos (x + y)
= 2 + 2 cos (x + y)
= 2 [1 + cos (x + y)]
= `2 xx 2 cos^2 ((x + y))/2 ...[because 1 + cos theta = 2 cos^2(theta/2)]`
= `4cos^2 ((x + y))/2`
= R.H.S.
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