1. **Problem Statement:** Prove the trigonometric identity $$\sin^2(x) + \cos^2(x) = 1$$. 2. **Formula and Rules:** This is a fundamental Pythagorean identity in trigonometry. It states that for any angle $x$, the square of the sine plus the square of the cosine equals 1. 3. **Proof:** - Consider a right triangle with hypotenuse length 1. - By definition, $\sin(x)$ is the ratio of the opposite side to the hypotenuse, and $\cos(x)$ is the ratio of the adjacent side to the hypotenuse. - Let the opposite side be $a$ and the adjacent side be $b$, so $\sin(x) = a/1 = a$ and $\cos(x) = b/1 = b$. - By the Pythagorean theorem, $a^2 + b^2 = 1^2 = 1$. - Substitute back: $\sin^2(x) + \cos^2(x) = a^2 + b^2 = 1$. 4. **Explanation:** This identity holds true for all angles $x$ because it is derived from the Pythagorean theorem applied to the unit circle or right triangles. **Final answer:** $$\sin^2(x) + \cos^2(x) = 1$$