Question

Prove the following:

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

Answer 1

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.