Simplify: [sin x cosy + cos x siny]2 + [cos x cosy - sin x siny]2

Here's how to simplify the expression: Step 1: Identify the trigonometric sum identities. The expression contains two parts that match the sum identities for sine and cosine: The first term, x y + x y, is the expansion of (x+y). The second term, x y - x y, is the expansion of (x+y). Step 2: Substitute the identities into the expression. [ x y + x y]^2 + [ x y - x y]^2 Becomes: [(x+y)]^2 + [(x+y)]^2 This can be written as: ^2(x+y) + ^2(x+y) Step 3: Apply the Pythagorean identity. Recall the fundamental trigonometric identity: ^2 + ^2 = 1. In this case, = (x+y). ^2(x+y) + ^2(x+y) = 1 The simplified expression is: 1 3 done, 2 left today. You're making progress.