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Prove It $\displaystyle \displaystyle \begin{align*} \csc{x} &= -2 \\ \frac{1}{\sin{x}} &= 2 \\ \sin{x} &= -\frac{1}{2} \end{align*}$
So the angle is in either the third or fourth quadrant.
$\displaystyle \displaystyle \begin{align*} \tan{x} &= \frac{\sqrt{3}}{3} \\ \tan{x} &= \frac{1}{\sqrt{3}} \end{align*}$
and since the tangent function is positive in the first and third quadrants, together, the two pieces of information tell us that the angle is in the third quadrant.
You should also know enough from the special triangles to be able to answer this question.