(cosx - cosy)² + (sinx - siny)² = 4 sin²(x-y)/2.

other mathod u can just .cosx-cosy
cosx-cosy= -2sin(x+y)/2. sin(x-y)/2 -----(1)

sinx-siny= 2 cos(x+y)/2 .sin(x-y)/2-----(2)

Therefore

(cosx-cosy)2+ (sinx-siny)2= 4sin2(x+y)/2 .sin2(x-y)/2 + 4cos2(x+y)/2.sin2(x-y)/2

= 4sin2(x-y)/2 {sin2(x+y)/2 + cos2(x+y)/2} =4sin2 (x-y)/2*1= 4sin2(x-y)/2 = RHs hence proved
 

thank you girish and guruprasad...

if rhs=4 cos square(x+y/2)

@girish - wats da difference between my solution and urs.. cheater

hahaha..u r right guruprasad..

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