1. The problem is to verify the trigonometric identity $\sin 2x = 2 \sin x \cos x$.\n\n2. Start with the double-angle formula for sine, which states that for any angle $x$,\n$$\sin 2x = 2 \sin x \cos x.$$\n\n3. This formula can be derived from the addition formula for sine: \n$$\sin (a + b) = \sin a \cos b + \cos a \sin b.$$\n\n4. Letting $a = b = x$, we have\n$$\sin 2x = \sin (x + x) = \sin x \cos x + \cos x \sin x = 2 \sin x \cos x.$$\n\n5. Hence, the identity $\sin 2x = 2 \sin x \cos x$ is true by definition and derivation from the sine addition formula.