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