Answer
$$\frac{{\sqrt {4 - {x^2}} }}{2}$$
Work Step by Step
$$\eqalign{
& \sin \left( {{{\cos }^{ - 1}}\left( {\frac{x}{2}} \right)} \right) \cr
& {\text{From the triangle shown below}} \cr
& \cos \theta = \frac{x}{2} \cr
& \theta = {\cos ^{ - 1}}\left( {\frac{x}{2}} \right),{\text{ then}} \cr
& \sin \left( {{{\cos }^{ - 1}}\left( {\frac{x}{2}} \right)} \right) = \sin \theta \cr
& {\text{From the triangle we obtain }}\sin \theta \cr
& \sin \left( {{{\cos }^{ - 1}}\left( {\frac{x}{2}} \right)} \right) = \frac{{\sqrt {4 - {x^2}} }}{2} \cr} $$
