The given expression is:

$(\sin x \cos y + \cos x \sin y)(\sin x \cos y - \cos x \sin y)$

Step 1: Recognize Trigonometric Identities

The terms inside the parentheses resemble the sine addition and sine subtraction identities:

1. Sum Identity: $\sin(x + y) = \sin x \cos y + \cos x \sin y$
2. Difference Identity: $\sin(x - y) = \sin x \cos y - \cos x \sin y$

Thus, we can rewrite the expression as:

$\sin(x + y) \cdot \sin(x - y)$

Step 2: Simplify Using Product-to-Sum Formula

We now use the product-to-sum identity for sine functions:

$\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]$

Substitute $A = x + y$ and $B = x - y$:

$\sin(x + y) \sin(x - y) = \frac{1}{2}[\cos((x + y) - (x - y)) - \cos((x + y) + (x - y))]$

Simplify the arguments:

$= \frac{1}{2}[\cos(2y) - \cos(2x)]$

Thus, the simplified expression is:

$\boxed{\frac{1}{2}[\cos(2y) - \cos(2x)]}$

Would you like further details or have any questions?

Related Questions:

1. How do we derive the sum and difference trigonometric identities?
2. Can this method be applied to other trigonometric functions?
3. How do you simplify trigonometric expressions using identities?
4. What is the product-to-sum formula for cosine functions?
5. How would this change if the original expression involved cosine instead of sine?

Tip:

Memorizing trigonometric identities is helpful, but understanding how to derive them is key to solving complex problems more intuitively.