LHS = (cos x – cos y)2 + (sin x – sin y)2
By expanding using formula
= cos2 x + cos2 y – 2 cos x cos y + sin2 x + sin2 y – 2 sin x sin y
Grouping the terms
= (cos2 x + sin2 x) + (cos2 y + sin2 y) – 2 (cos x cos y + sin x sin y)
Using the formula cos (A – B) = cos A cos B + sin A sin B
= 1 + 1 – 2 [cos (x – y)]
By further calculation
= 2 [1 – cos (x – y)]
From formula cos 2A = 1 – 2 sin2 A
\( 2[1-\{1-2sin^2(\dfrac{x-y}{2})\}]\)
we get,
= \( 4sin^2(\dfrac{x-y}{2})\)
Therefore L.H.S = R.H.S
Hence proved.
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