Answer
The solution set is $$\{60^\circ+180^\circ n, n\in Z\}$$
Work Step by Step
$$\tan\theta\csc\theta-\sqrt3\csc\theta=0$$
1) Solve the equation over the interval $[0^\circ,360^\circ)$
$$\tan\theta\csc\theta-\sqrt3\csc\theta=0$$
$$\csc\theta(\tan\theta-\sqrt3)=0$$
$$\csc\theta=0\hspace{1cm}\text{or}\hspace{1cm}\tan\theta=\sqrt3$$
- For $\csc\theta=0$
The range of a cosectant function is $(-\infty,-1]\cup[1,\infty)$. Since $0\notin(-\infty,-1]\cup[1,\infty)$, there cannot be any values of $\theta\in[0^\circ, 360^\circ)$ that have $\csc\theta=0$.
- For $\tan\theta=\sqrt3$
Over the interval $[0^\circ, 360^\circ)$, there are 2 values of $\theta$ where $\tan\theta=\sqrt3$, which are $60^\circ$ and $240^\circ$.
Therefore, overall, $$\theta=\{60^\circ, 240^\circ\}$$
2) Solve the equation for all solutions
- The integer multiples of the period of the tangent function is $180^\circ$.
- We apply it to each solution found in step 1, which would lead to $\theta=60^\circ+180^\circ n$ and $\theta=240^\circ+180^\circ n$ $(n\in Z)$
- However, both $60^\circ+180^\circ n$ and $240^\circ+180^\circ n$ refer to the same set of points, so we only need to include one in the solution set.
The solution set is $$\{60^\circ+180^\circ n, n\in Z\}$$
