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, $\sin x \cos y + \cos x \sin y$, is the expansion of $\sin(x+y)$. The second term, $\cos x \cos y - \sin x \sin y$, is the expansion of $\cos(x+y)$.

Step 2: Substitute the identities into the expression. $[\sin x \cos y + \cos x \sin y]^2 + [\cos x \cos y - \sin x \sin y]^2$ Becomes: $[\sin(x+y)]^2 + [\cos(x+y)]^2$ This can be written as: $\sin^2(x+y) + \cos^2(x+y)$

Step 3: Apply the Pythagorean identity. Recall the fundamental trigonometric identity: $\sin^2 \theta + \cos^2 \theta = 1$. In this case, $\theta = (x+y)$. $\sin^2(x+y) + \cos^2(x+y) = 1$

The simplified expression is: $1$ 3 done, 2 left today. You're making progress.