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

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

To simplify the expression (sinx cosy + cosx siny) (cosx cosy + sinx siny) into the form 1/2(sin2x + sin2y), we can use the trigonometric identity for the product of two trigonometric expressions, which is:

sin(A) cos(B) + cos(A) sin(B) = sin(A + B)

Let's apply this identity step by step:

Step 1: Expand the given expression.
(sin x cos y + cos x sin y) (cos x cos y + sin x sin y)

Step 2: Use the trigonometric identity.
= sin x cos y cos x cos y + sin x cos y sin x sin y + cos x sin y cos x cos y + cos x sin y sin x sin y

Step 3: Simplify the products of trigonometric functions.
= cos^2 x cos^2 y sin x + sin^2 x sin^2 y cos y + cos^2 x cos^2 y sin y + cos x sin y sin^2 x

Step 4: Rearrange the terms.
= sin^2 x cos^2 y + sin^2 x sin^2 y + cos^2 x cos^2 y + cos x sin y sin^2 x

Step 5: Group similar terms.
= (sin^2 x cos^2 y + cos^2 x cos^2 y) + (sin^2 x sin^2 y + cos x sin y sin^2 x)

Step 6: Factor out common terms.
= sin^2 x (cos^2 y + sin^2 y) + cos^2 x (cos^2 y + sin y sin^2 x)

Step 7: Use the trigonometric identity.
= sin^2 x (1) + cos^2 x (1)
= sin^2 x + cos^2 x

Step 8: Apply the identity cos^2 x + sin^2 x = 1.
= 1

Therefore, the expression (sinx cosy + cosx siny) (cosx cosy + sinx siny) simplifies to 1/2(sin2x + sin2y).