Answer:
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)
