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\frac1{\sin x}=-2\iff\sin x=-\frac12`

`\implies x=\begin{cases}\frac{7\pi}6+2n\pi\\\\\frac{11\pi}6+2n\pi\end{cases}`


`\csc x+1=0\iff\frac1{\sin x}=-1\iff\sin x=-1`

`\implies x=\frac{3\pi}2+2n\pi`

In all solutions,
`n` is any integer.