We simplify the given expression:

$\frac{\sin x}{-\sin(2x)}$

Step 1: Use the Double-Angle Identity

We know that:

$\sin(2x) = 2\sin x \cos x$

Substituting this in:

$\frac{\sin x}{- (2\sin x \cos x)}$

Step 2: Cancel Common Terms

Since $\sin x$ appears in both the numerator and denominator (as long as $\sin x \neq 0$):

$\frac{1}{-2\cos x}$

which simplifies to:

$-\frac{1}{2\cos x}$

Final Answer:

$-\frac{1}{2\cos x}$

Would you like a step-by-step breakdown for better understanding?

Here are some related questions to explore:

1. What happens if $x = 0$ in the simplified expression?
2. How does this expression behave as $x$ approaches $\frac{\pi}{2}$?
3. Can we rewrite this in terms of tangent or cotangent?
4. How does the denominator affect the domain of the function?
5. What is the significance of the negative sign in the result?

Tip: Always check for domain restrictions when simplifying trigonometric expressions! 🚀