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Question
Prove that: `(cos x - cosy)^2 + (sin x - sin y)^2 = 4 sin^2 (x - y)/2`
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Solution
L.H.S. = (cos x – cos y)2 + (sin x – sin y)2
= `( -2sin (x + y)/2 sin (x - y)/2)^2 + (2cos (x + y)/2 sin (x - y)/2)^2`
= `4sin^2 (x +y)/2 sin^2 (x - y)/2 + 4cos^2 (x +y)/2 sin^2 (x - y)/2`
= `4sin^2 (x -y) [ sin^2 (x + y)/2 + cos^2 (x +y)/2]`
= `4sin^2 (x - y)/2` `[∵ sin^2 (x + y)/2 + cos^2 (x +y)/2 = 1]`
= R.H.S.
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