1. **State the problem:** Simplify the expression $\sin(x + \Delta x) - \sin x$. 2. **Use the sine subtraction formula:** Recall the identity for the difference of sines: $$\sin A - \sin B = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right)$$ 3. **Apply the formula:** Let $A = x + \Delta x$ and $B = x$, then $$\sin(x + \Delta x) - \sin x = 2 \cos \left( \frac{(x + \Delta x) + x}{2} \right) \sin \left( \frac{(x + \Delta x) - x}{2} \right)$$ 4. **Simplify inside the cosine and sine:** $$= 2 \cos \left( \frac{2x + \Delta x}{2} \right) \sin \left( \frac{\Delta x}{2} \right) = 2 \cos \left( x + \frac{\Delta x}{2} \right) \sin \left( \frac{\Delta x}{2} \right)$$ 5. **Final simplified form:** $$\sin(x + \Delta x) - \sin x = 2 \cos \left( x + \frac{\Delta x}{2} \right) \sin \left( \frac{\Delta x}{2} \right)$$ This expression shows the difference of sine values as a product of cosine and sine functions, which is useful in calculus and trigonometry for understanding changes in sine values.