Which value equals sin^2 θ + cos^2 θ for any angle θ?

• A. 1
• B. sin θ
• C. cos θ
• D. sin^2 θ − cos^2 θ

Answer: (A)

The essential idea is the Pythagorean identity: for any angle, the squares of sine and cosine add up to 1. On the unit circle, a point is (cos θ, sin θ), and its distance from the origin is 1, so cos^2 θ + sin^2 θ = 1. You can also see it from a right triangle: sin θ = opposite/hypotenuse and cos θ = adjacent/hypotenuse, so sin^2 θ + cos^2 θ = (opposite^2 + adjacent^2)/hypotenuse^2 = hypotenuse^2/hypotenuse^2 = 1. This is constant, regardless of θ. The other expressions depend on θ: sin θ and cos θ vary, and sin^2 θ − cos^2 θ equals cos 2θ, which changes with θ, so none of them stay identically 1.