For real angles x and y, what is the simplified value of the expression [(sin x + sin y)(sin x − sin y)] / [(cos x + cos y)(cos y − cos x)]?

Introduction / Context:
This problem tests simplification of a trigonometric expression using basic identities. The expression involves sums and differences of sines and cosines in product form, and the goal is to simplify it to a constant, if possible, without assuming any special values of x or y.

Given Data / Assumptions:
- Expression: [(sin x + sin y)(sin x − sin y)] / [(cos x + cos y)(cos y − cos x)].
- x and y are real angles for which all trigonometric functions involved are defined.

Concept / Approach:
We recognize that (a + b)(a − b) = a² − b². The numerator is a difference of squares in sine terms, and the denominator is a difference of squares in cosine terms, except for the sign hidden in (cos y − cos x). By rewriting both numerator and denominator using this pattern and Pythagorean identities, we can see whether the expression reduces to a simple constant.

Step-by-Step Solution:
Step 1: Simplify the numerator: (sin x + sin y)(sin x − sin y) = sin²x − sin²y.Step 2: Simplify the denominator: (cos x + cos y)(cos y − cos x) = cos y² − cos x² = cos²y − cos²x.Step 3: Use the identity sin²θ = 1 − cos²θ.Step 4: Compute sin²x − sin²y = (1 − cos²x) − (1 − cos²y) = −cos²x + cos²y = cos²y − cos²x.Step 5: Notice that the numerator is exactly cos²y − cos²x, which matches the denominator.Step 6: Therefore, the expression equals (cos²y − cos²x) / (cos²y − cos²x) = 1, as long as the denominator is not zero.

Verification / Alternative check:
You can verify with sample values, for example x = 30° and y = 60°. Substituting these into the expression and simplifying numerically will give a value extremely close to 1, confirming the algebraic simplification.

Why Other Options Are Wrong:
The values 0, −1 and 2 do not match the algebraic simplification. Getting 0 often comes from mistakenly cancelling terms incorrectly. Obtaining −1 or 2 typically results from sign errors in handling cos y − cos x or misuse of the Pythagorean identity.

Common Pitfalls:
A common mistake is to forget that (cos y − cos x) = −(cos x − cos y) and mishandle the sign, which changes the result. Another pitfall is mixing up sin²θ + cos²θ = 1 with incorrect variants like sin²θ − cos²θ = 1, which is not true. Always apply algebraic identities carefully and match patterns like a² − b² correctly.

Final Answer:
The simplified value of the expression is 1.