Answer
The solution set to this problem is $$\{45^\circ,225^\circ\}$$
Work Step by Step
$$\csc^2\theta-2\cot\theta=0$$ over interval $[0^\circ,360^\circ)$
1) Solve the equation:
$$\csc^2\theta-2\cot\theta=0$$
Here we have two different trigonometric functions: cosecant and cotangent. It would be better if we have only one type, since only by then can we solve trigonometric functions.
Seeing that we have $\csc^2\theta$, we can recall the identity: $\csc^2\theta=1+\cot^2\theta$. In fact, we would replace $\csc^2\theta$ with $1+\cot^2\theta$
$$1+\cot^2\theta-2\cot\theta=0$$
$$(\cot\theta-1)^2=0$$
$$\cot\theta-1=0$$
$$\cot\theta=1$$
2) Apply the inverse function:
Over the interval $[0^\circ,360^\circ)$, $\cot\theta\gt0$ means that $\theta$ angle lies either in quadrant I or quadrant III. In quadrant I, angle $45^\circ$ and in quadrant III, angle $225^\circ$ would have cosecant equal $1$.
Therefore, $$\theta\in\{45^\circ,225^\circ\}$$
In other words, the solution set to this problem is $$\{45^\circ,225^\circ\}$$
