Solve the equation for solutions in the interval $[0,2\pi )$ by first solving for the trigonometric function.
$$(\cot x-1)(\sqrt{3}\tan x-1)=0$$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is (Simplify your answer. Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
B. The solution set is the empty set.

Step 1 ：The given equation is a product of two factors equal to zero. This means that either of the factors can be zero. So, we can solve the equation by setting each factor equal to zero and solving for x. The solutions to the equation will be the union of the solutions to each of these equations.

Step 2 ：The first equation is $\cot x-1=0$, which simplifies to $\cot x=1$. The solutions to this equation in the interval $[0,2\pi )$ are $x=\frac{\pi }{4}$ and $x=\frac{5\pi }{4}$.

Step 3 ：The second equation is $\sqrt{3}\tan x-1=0$, which simplifies to $\tan x=\frac{1}{\sqrt{3}}$. The solutions to this equation in the interval $[0,2\pi )$ are $x=\frac{\pi }{6}$ and $x=\frac{7\pi }{6}$.

Step 4 ：So, the solution set to the given equation is $\{\frac{\pi }{4},\frac{5\pi }{4},\frac{\pi }{6},\frac{7\pi }{6}\}$.

Step 5 ：Final Answer: The solution set is $\boxed{{\displaystyle \{\frac{\pi }{4},\frac{5\pi }{4},\frac{\pi }{6},\frac{7\pi }{6}\}}}$.