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

Explanation

`L.H.S= ( cos x + cos y)^(2) + ( sin x - siny)^(2)`
` = [2 cos ((x+y)/2) cos.((x-y)/2)]^(2)`
` + [ 2 cos ((x+y)/2)sin.((x-y)/2)]^(2)`
` 4 cos^(2) ((x+y)/2) cos^(2)((x-y)/2)`
` + 4 cos^(2) ((x+y)/2)sin^(2) ((x-y)/2)`
` = 4 cos^(2) ((x+y)/2) [ cos^(2) ((x+y)/2) + sin^(2) ((x-y)/2)]`
` = 4 cos^(2) ((x+y)/2) (1)`
` = R.H.S`