\(\cot^2x+3\csc x=-3\)
\(\csc^2x-1+3\csc x+3=0\)
\(\csc^2x+3\csc x+2=0\)
\((\csc x+2)(\csc x+1)=0\)
\(\csc x+2=0\iff\dfrac1{\sin x}=-2\iff\sin x=-\dfrac12\)
\(\implies x=\begin{cases}\dfrac{7\pi}6+2n\pi\\\\\dfrac{11\pi}6+2n\pi\end{cases}\)
\(\csc x+1=0\iff\dfrac1{\sin x}=-1\iff\sin x=-1\)
\(\implies x=\dfrac{3\pi}2+2n\pi\)
In all solutions, \(n\) is any integer.
