How do I get (sinx cosy + cosx siny) (cosx cosy + sinx siny) in the form of 1/2(sin2x + sin2y)?

Answer 1

sinx cosy + cosx siny = sin(x+y)
cosx cosy + sinx siny = cos(x-y)

using the product-to-sum formulas,

sinA cosB = 1/2 (sin(A+B)+sin(A-B))

so, just plug in
A = x+y
B = x-y