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

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 $\cos(x+y)$.

Becomes:

This can be written as:

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

The simplified expression is: