1. The problem is to prove the trigonometric identity $$\sin^2(x) + \cos^2(x) = 1$$. 2. This identity is fundamental in trigonometry and comes from the Pythagorean theorem applied to a right triangle inscribed in the unit circle. 3. Consider a point on the unit circle at an angle $x$ from the positive x-axis. The coordinates of this point are $(\cos(x), \sin(x))$. 4. By the definition of the unit circle, the radius is 1, so the distance from the origin to this point satisfies: $$\cos^2(x) + \sin^2(x) = 1^2 = 1$$ 5. This is exactly the identity we wanted to prove. 6. Therefore, $$\sin^2(x) + \cos^2(x) = 1$$ holds for all real numbers $x$.