Which of the following transformations relates sin^2(x) + cos^2(x)?

The statement that sin²(x) + cos²(x) always equals 1 is a fundamental trigonometric identity known as the Pythagorean identity. This identity holds true for all values of x and is derived from the definition of sine and cosine on the unit circle.

To understand why this identity is true, consider a point on the unit circle, defined by an angle x. The x-coordinate of that point corresponds to cos(x), and the y-coordinate corresponds to sin(x). By the Pythagorean theorem, the distance from the origin to any point on the unit circle equals 1. Therefore, when you square both coordinates and add them together, the sum will always be equal to 1, consistent with the equation sin²(x) + cos²(x) = 1.

This identity applies universally and reflects the inherent relationship between sine and cosine as they relate to circular functions. Hence, it remains a cornerstone of trigonometric equations and problem-solving.

The other alternatives provided either misrepresent the relationship between sine and cosine or suggest variability that contradicts the established identity.