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