Proving sin²(x) + cos²(x) = 1

To prove the identity sin²(x) + cos²(x) = 1, we start with the Pythagorean identity for the unit circle. By considering a right-angled triangle with hypotenuse of length 1, and angles x, sin(x) = opposite/hypotenuse and cos(x) = adjacent/hypotenuse. Using the Pythagorean theorem, we find that sin²(x) + cos²(x) = 1. This fundamental trigonometric identity is crucial in various mathematical applications, especially in calculus and physics.

In conclusion, sin²(x) + cos²(x) = 1 is a fundamental identity in trigonometry, representing the relationship between sine and cosine functions on the unit circle.