1. **State the problem:** Simplify the expression $\sin\left(\frac{3\pi}{2} - x\right)$. 2. **Recall the sine difference identity:** For any angles $A$ and $B$, $$\sin(A - B) = \sin A \cos B - \cos A \sin B.$$ 3. **Apply the identity:** Let $A = \frac{3\pi}{2}$ and $B = x$, so $$\sin\left(\frac{3\pi}{2} - x\right) = \sin\frac{3\pi}{2} \cos x - \cos\frac{3\pi}{2} \sin x.$$ 4. **Evaluate the trigonometric values:** - $\sin\frac{3\pi}{2} = -1$ - $\cos\frac{3\pi}{2} = 0$ 5. **Substitute these values back:** $$\sin\left(\frac{3\pi}{2} - x\right) = (-1) \cdot \cos x - 0 \cdot \sin x = -\cos x.$$ 6. **Final simplified form:** $$\sin\left(\frac{3\pi}{2} - x\right) = -\cos x.$$ **Answer:** d. $-\cos x$