Answer
$$x = \frac{1}{3}$$
Work Step by Step
$$\eqalign{
& \arcsin \sqrt {2x} = \arccos \sqrt x \cr
& {\text{sin}}\left( {\arcsin \sqrt {2x} } \right) = \sin \left( {\arccos \sqrt x } \right) \cr
& \cr
& {\text{*From the triangle shown below, we obtain }} \cr
& \sin \left( {\arccos \sqrt x } \right) = \frac{{\sqrt {1 - x} }}{1} = \sqrt {1 - x} \cr
& \cr
& \underbrace {{\text{sin}}\left( {\arcsin \sqrt {2x} } \right) = \sin \left( {\arccos \sqrt x } \right)}_ \downarrow \cr
& \sqrt {2x} = \sqrt {1 - x} ,{\text{ for }}0 \leqslant x \leqslant 1 \cr
& {\text{Square both sides}} \cr
& 2x = 1 - x \cr
& 3x = 1 \cr
& x = \frac{1}{3} \cr} $$
