1. **Problem Statement:** Find the difference between $\sin 2x$ and $2 \sin x$. 2. **Recall the formula for double angle sine:** $$\sin 2x = 2 \sin x \cos x$$ 3. **Compare $\sin 2x$ and $2 \sin x$:** - $\sin 2x = 2 \sin x \cos x$ - $2 \sin x$ is just twice the sine of $x$ without the cosine factor. 4. **Difference:** $$\sin 2x - 2 \sin x = 2 \sin x \cos x - 2 \sin x = 2 \sin x (\cos x - 1)$$ 5. **Interpretation:** - The difference depends on $\cos x - 1$. - Since $\cos x$ ranges from $-1$ to $1$, $\cos x - 1$ ranges from $-2$ to $0$. - Therefore, $\sin 2x$ equals $2 \sin x$ only when $\cos x = 1$, i.e., when $x = 2k\pi$ for integers $k$. **Final answer:** $$\sin 2x - 2 \sin x = 2 \sin x (\cos x - 1)$$