1. The problem is to prove that $\sin^2 x + \cos^2 x = 1$ for any angle $x$. 2. Recall the Pythagorean identity from trigonometry which states that the square of the sine of an angle plus the square of the cosine of the same angle equals one. 3. This identity can be derived from the Pythagorean theorem applied to a right triangle on the unit circle. 4. On the unit circle, any point is represented as $(\cos x, \sin x)$ where $x$ is the angle formed with the positive x-axis. 5. The distance from the origin to any point on the unit circle is 1, so by the Pythagorean theorem: $$\cos^2 x + \sin^2 x = 1^2 = 1$$ 6. Thus, the identity $\sin^2 x + \cos^2 x = 1$ is proven.