Question

Cot^2x + 3cscx=-3 all real numbers solve for x

Answer 1

$\cot^2x+3\csc x=-3$
$\csc^2x-1+3\csc x+3=0$
$\csc^2x+3\csc x+2=0$
$(\csc x+2)(\csc x+1)=0$

$\csc x+2=0\iff\dfrac1{\sin x}=-2\iff\sin x=-\dfrac12$
$\implies x=\begin{cases}\dfrac{7\pi}6+2n\pi\\\\\dfrac{11\pi}6+2n\pi\end{cases}$

$\csc x+1=0\iff\dfrac1{\sin x}=-1\iff\sin x=-1$
$\implies x=\dfrac{3\pi}2+2n\pi$

In all solutions, $n$ is any integer.