Maharashtra State BoardHSC Science (General) 11th
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Prove the following: (cos x – cos y)2 +(sin x – sin y)2 = 4sin2 (x-y)2 - Mathematics and Statistics

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Sum

Prove the following:

(cos x – cos y)2 + (sin x – sin y)2 = `4sin^2  ((x - y))/2`

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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.

Concept: Trigonometric Functions of Triple Angle
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