1. The problem is to verify the trigonometric identity: $\cos 2x = 1 - 2 \sin^2 x$. 2. The double-angle formula for cosine states that: $$\cos 2x = \cos^2 x - \sin^2 x$$ 3. Using the Pythagorean identity: $$\cos^2 x = 1 - \sin^2 x$$ 4. Substitute $\cos^2 x$ in the double-angle formula: $$\cos 2x = (1 - \sin^2 x) - \sin^2 x$$ 5. Simplify the right side: $$\cos 2x = 1 - \sin^2 x - \sin^2 x = 1 - 2 \sin^2 x$$ 6. This matches the given identity, so it is verified. Therefore, the identity $\cos 2x = 1 - 2 \sin^2 x$ is true.