[tex]\bf 2[sin(x)]^2+\sqrt{3}sin(x)=2sin(x)+\sqrt{3}
\\\\\\
2[sin(x)]^2+\sqrt{3}sin(x)-2sin(x)-\sqrt{3}=0
\\\\\\\
[2sin^2(x)-2sin(x)]~+~[\sqrt{3}sin(x)-\sqrt{3}]=0
\\\\\\
\stackrel{common~factor}{2sin(x)}[sin(x)-1]~~+~~\stackrel{common~factor}{\sqrt{3}}[sin(x)-1]=0
\\\\\\[/tex]
[tex]\bf \stackrel{common~factor}{[sin(x)-1]}~[2sin(x)+\sqrt{3}]=0\qquad
\begin{cases}
sin(x)-1=0\\
sin(x)=1\\
\measuredangle x=\frac{\pi }{2}\\
----------\\
2sin(x)+\sqrt{3}=0\\
2sin(x)=-\sqrt{3}\\
sin(x)=-\frac{\sqrt{3}}{2}\\\\
\measuredangle x=\frac{4\pi }{3}~,~\frac{5\pi }{3}
\end{cases}[/tex]
