1. The problem is to simplify $\sin\left(\frac{3\pi}{2} - x\right)$. 2. We use the sine difference identity: $$\sin(a - b) = \sin a \cos b - \cos a \sin b$$ 3. Substitute $a = \frac{3\pi}{2}$ and $b = x$: $$\sin\left(\frac{3\pi}{2} - x\right) = \sin\frac{3\pi}{2} \cos x - \cos\frac{3\pi}{2} \sin x$$ 4. Evaluate $\sin\frac{3\pi}{2}$ and $\cos\frac{3\pi}{2}$: - $\sin\frac{3\pi}{2} = -1$ - $\cos\frac{3\pi}{2} = 0$ 5. Substitute these values back: $$-1 \cdot \cos x - 0 \cdot \sin x = -\cos x$$ 6. Therefore, the simplified form is $-\cos x$. 7. The correct answer is option d. $-\cos x$.