Here you go:
1) Applying (a + b)² = a² + b² + 2ab,
(cosx + cosy)² = cos²x + cos²y + 2cos(x)*cos(y) and
(sinx + siny)² = sin²x + sin²y + 2sin(x)*sin(y)
2) ==> (cosx + cosy)² + (sinx + siny)² =
= (cos²x + sin²x) + (cos²y + sin²y) + 2{cos(x)cos(y) + sin(x)sin(y)}
= 2 + 2{cos(x - y)} = 2[1 + cos(x - y)]
= 2*2cos²{(x - y)/2} [Multiple angle identity, 1 + cos(2A) = 2cos²A]
= 4*cos²{(x - y)/2} [Proved]
