Answer
See the steps.
Work Step by Step
$\sin{(\dfrac{3\pi}{2}+ x)} + \sin{(\dfrac{3\pi}{2}- x)} =$
$$\sin{\dfrac{3\pi}{2}} \cos{x} + \cos{\dfrac{3\pi}{2}} \sin{x} + \sin{\dfrac{3\pi}{2}} \cos{x}- \cos{\dfrac{3\pi}{2}} \sin{x}$$
$LHS = 2 \sin{\dfrac{3\pi}{2}} \cos{x} = 2 \times -1 \cos{x} = -2 \cos{x}$
$LHS = RHS$
