(cosx - cosy)^{2} + (sinx - siny)^{2} = 4 sin^{2}(x-y)/2.

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}= 4sin^{2}(x+y)/2 .sin^{2}(x-y)/2 + 4cos^{2}(x+y)/2.sin^{2}(x-y)/2

= 4sin^{2}(x-y)/2 {sin^{2}(x+y)/2 + cos^{2}(x+y)/2} =4sin^{2 }(x-y)/2*1= 4sin^{2}(x-y)/2 = RHs hence proved