Final answer:
The expression sin^2 x - cos^2 x over (sin x - cos x)^2 simplifies to sin x + cos x by recognizing the numerator as a difference of squares and cancelling one (sin x - cos x) term with the denominator.
Step-by-step explanation:
The question is about simplifying a trigonometric expression by factoring the denominator and then simplifying it. We are presented with the expression sin2 x - cos2 x over (sin x - cos x)2 which we need to simplify.
To solve this, we first recognize that the numerator is a form of the difference of squares, which can be written as (sin x + cos x)(sin x - cos x). Since the denominator is (sin x - cos x)2, we can simplify the expression by cancelling out one (sin x - cos x) term. This yields sin x + cos x after simplification.
Further insights can be obtained by exploring trigonometric identities such as the Pythagorean identity or double angle formulas, but this is not necessary for this specific simplification.