1. We start with the problem: simplify or express $\sin 2x$ in terms of $\sin x$ and $\cos x$. 2. The double-angle formula for sine is: $$\sin 2x = 2 \sin x \cos x$$ This formula is very useful because it expresses the sine of a double angle in terms of the sine and cosine of the original angle. 3. Important rule: The sine of twice an angle is twice the product of sine and cosine of that angle. 4. So, if you have $\sin 2x$, you can rewrite it as: $$\sin 2x = 2 \sin x \cos x$$ 5. This is the simplest and most common form for $\sin 2x$. 6. If you want to evaluate $\sin 2x$ for a specific value of $x$, just plug in the values of $\sin x$ and $\cos x$ and multiply by 2. Final answer: $$\sin 2x = 2 \sin x \cos x$$