Answer
$\theta = 135^{\circ}~~$ or $~~\theta = 315^{\circ}$
Work Step by Step
$csc^2~\theta - 2~cot~\theta = 4$
$\frac{1}{sin^2~\theta} - \frac{2~cos~\theta}{sin~\theta} = 4$
$\frac{1}{sin^2~\theta} - \frac{2~sin~\theta~cos~\theta}{sin^2~\theta} = 4$
$1 - 2~sin~\theta~cos~\theta = 4~sin^2~\theta$
$(sin^2~\theta+cos^2~\theta) - 2~sin~\theta~cos~\theta = 4~sin^2~\theta$
$(sin~\theta-cos~\theta)^2 = 4~sin^2~\theta$
$sin~\theta-cos~\theta = 2~sin~\theta$
$-cos~\theta = sin~\theta$
$tan~\theta = -1$
$\theta = 135^{\circ}~~$ or $~~\theta = 315^{\circ}$
