Which statement represents the Pythagorean identity?

• A. cos^2 x + sin^2 x = 2
• B. sin x + cos x = 1
• C. sin^2 x + cos^2 x = 1
• D. sin^2 x - cos^2 x = 0

Answer: (C)

The Pythagorean identity expresses a universal relationship between sine and cosine: sin^2 x + cos^2 x equals 1 for any angle x. This comes from the unit circle, where a point on the circle has coordinates (cos x, sin x) and satisfies cos^2 x + sin^2 x = 1.

So the statement sin^2 x + cos^2 x = 1 matches this fundamental relation exactly, making it the correct representation.

The other forms don’t hold for all angles. cos^2 x + sin^2 x = 2 would force the sum to be 2, which contradicts the unit-circle relation. sin x + cos x = 1 is not true for all x; it would only be true for specific angles. sin^2 x − cos^2 x = 0 would only occur at angles where sin^2 x equals cos^2 x, not universally.