Which identity represents the Pythagorean theorem for sine and cosine?

The correct identity representing the Pythagorean theorem for sine and cosine is sin² + cos² = 1. This fundamental relationship illustrates how the square of the sine of an angle and the square of the cosine of the same angle sum to one. It derives directly from the geometry of the unit circle, where any point on the circle can be represented in terms of the sine and cosine of an angle.

In the context of the unit circle, the coordinates of any point are given by (cos(θ), sin(θ)). Since any point on the unit circle is at a distance of 1 from the origin, the equation of the circle can be expressed as x² + y² = 1. Substituting x with cos(θ) and y with sin(θ), we arrive at the identity sin²(θ) + cos²(θ) = 1.

Understanding this identity is crucial in trigonometry and has applications in various fields of mathematics, engineering, and physics. It serves as a foundation for proving other trigonometric identities and equations.