Answer
$$-\cot \left(x\right)$$
Work Step by Step
Simplifying the expression using the sum and difference formulas, we find:
$$\frac{\cos \left(\frac{3\pi }{2}\right)\sin \left(x\right)+\cos \left(x\right)\sin \left(\frac{3\pi }{2}\right)}{\cos \left(\frac{3\pi }{2}+x\right)} \\ -\frac{\cos \left(x\right)}{\sin \left(x\right)} \\ -\cot \left(x\right)$$
