$\begin{array}{l}
2{\sin ^2}x - 3\sin x + 1 = 0\\
\Leftrightarrow \left( {2\sin x - 1} \right)\left( {\sin x - 1} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\sin x = \dfrac{1}{2}\\
\sin x = 1
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{\pi }{6} + k2\pi \\
x = \dfrac{{5\pi }}{6} + k2\pi \\
x = \dfrac{\pi }{2} + k2\pi
\end{array} \right.\\
k = 0 \Rightarrow x = \dfrac{\pi }{6},x = \dfrac{{5\pi }}{6},x = \dfrac{\pi }{2}\\
\Rightarrow x = \dfrac{\pi }{6} \in \left( {0;\dfrac{\pi }{2}} \right)\\
k = - 1 = x = \dfrac{{ - 11\pi }}{6},x = \dfrac{{ - 7\pi }}{6},x = \dfrac{{ - 3\pi }}{2} \notin \left( {0;\dfrac{\pi }{2}} \right)\\
\Rightarrow x = \dfrac{\pi }{6}
\end{array}$
