$2\sin^2\alpha-3\sin\alpha-2=0$
$\Leftrightarrow (\sin \alpha-2)(2\sin\alpha+1)=0$
$\Leftrightarrow \sin \alpha=\dfrac{-1}{2}$
$\dfrac{1}{\sin^2\alpha}=1+\cot^2\alpha$
$\Rightarrow \cot^2\alpha=3$
$\Leftrightarrow \tan^2\alpha=\dfrac{1}{3}$
$\Leftrightarrow \tan \alpha=\pm\dfrac{1}{\sqrt3}$