Answer
$$x=\frac{4\pi }{3}+2\pi n,\:x=\frac{5\pi }{3}+2\pi n$$
Work Step by Step
We solve the equation using the properties of trigonometric functions. Note, there is a general solution since trigonometric identities go up and down and this can pass through a given value of y many times. Solving this, we find:
$$\sqrt{3}\csc \left(x\right)=-2 \\ \csc \left(x\right)=-\frac{2\sqrt{3}}{3} \\\ x=\frac{4\pi }{3}+2\pi n,\:x=\frac{5\pi }{3}+2\pi n$$
