1. **State the problem:** Simplify the expression $\sin(x + y) - \sin(x - y)$. 2. **Recall the sine subtraction formula:** For any angles $A$ and $B$, $$\sin A - \sin B = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right)$$ 3. **Apply the formula:** Let $A = x + y$ and $B = x - y$. Then, $$\sin(x + y) - \sin(x - y) = 2 \cos \left( \frac{(x + y) + (x - y)}{2} \right) \sin \left( \frac{(x + y) - (x - y)}{2} \right)$$ 4. **Simplify inside the cosine and sine:** $$\frac{(x + y) + (x - y)}{2} = \frac{2x}{2} = x$$ $$\frac{(x + y) - (x - y)}{2} = \frac{2y}{2} = y$$ 5. **Substitute back:** $$\sin(x + y) - \sin(x - y) = 2 \cos(x) \sin(y)$$ 6. **Final answer:** $$\boxed{2 \cos(x) \sin(y)}$$