Prove the identity
(cosx+cosy)^2+(sinx-siny)^2=2+2co s(x+y)​

See below.

Step-by-step explanation:

(cos x + cos y)² + (sin x - sin y)²

       Expand each squared term

= (cos x + cos y)*(cos x + cos y) + (sin x - sin y)*(sin x - sin y)

= (cos²x + 2 cos x cos y + cos²y) + (sin²x - 2 sin x siny + sin²y)

       Simplify the expression

= (cos²x + sin²x) + (cos²y + sin²y) + 2(cos x cos y - sin x sin y)

       Apply the Pythagorean identity: (sin²x + cos²x = 1)

= 1 + 1 + 2(cos x cos y - sin x sin y)

       Apply the addition formula for cosine: cos(x+y) = cos x cos y - sin x sin y

= 1 + 1 + 2cos(x+y)

       Add together everything

= 2 + 2cos(x+y)