Answer
The required solution is $\frac{\sqrt{3}}{2}$
Work Step by Step
Let,
$\begin{align}
& {{\sin }^{-1}}\frac{\sqrt{3}}{2}=x \\
& \sin x=\frac{\sqrt{3}}{2} \\
\end{align}$
The value of $\sin \frac{\pi }{3}$ is $\frac{\sqrt{3}}{2}$. Thus,
$\begin{align}
& \sin x=\sin \frac{\pi }{3}\text{ } \\
& x=\frac{\pi }{3}
\end{align}$
The value of ${{\sin }^{-1}}\frac{\sqrt{3}}{2}=x$, and $x=\frac{\pi }{3}$ . Therefore, the value of ${{\sin }^{-1}}\frac{\sqrt{3}}{2}$ is $\frac{\pi }{3}$.
Thus, the expression is further simplified by putting the value of ${{\sin }^{-1}}\frac{\sqrt{3}}{2}$.
$\begin{align}
& \sin \left( 2{{\sin }^{-1}}\frac{\sqrt{3}}{2} \right)=\sin \left( 2\cdot \frac{\pi }{3} \right) \\
& =\sin \left( \frac{2\pi }{3} \right)\text{ } \\
& =\frac{\sqrt{3}}{2}
\end{align}$
