Answer
The solution set to this problem is $$\{\frac{\pi}{4},\frac{2\pi}{3},\frac{5\pi}{4},\frac{5\pi}{3}\}$$
Work Step by Step
$$(\cot x-1)(\sqrt3\cot x+1)=0$$ over interval $[0,2\pi)$
1) Solve the equation:
$$(\cot x-1)(\sqrt3\cot x+1)=0$$
$$\cot x-1=0\hspace{1cm}\text{or}\hspace{1cm}\sqrt3\cot x+1=0$$
$$\cot x=1\hspace{1cm}\text{or}\hspace{1cm}\cot x=-\frac{1}{\sqrt3}=-\frac{\sqrt3}{3}$$
2) Apply the inverse function:
- $\cot x=1$: Over the interval $[0,2\pi)$, there are 2 values whose cotangent equals $1$, which are $\frac{\pi}{4}$ and $\frac{5\pi}{4}$
- $\cot x=-\frac{\sqrt3}{3}$: Over the interval $[0,2\pi)$, there are 2 values whose cotangent equals $-\frac{\sqrt3}{3}$, which are $\frac{2\pi}{3}$ and $\frac{5\pi}{3}$
Therefore, $$x\in\{\frac{\pi}{4},\frac{2\pi}{3},\frac{5\pi}{4},\frac{5\pi}{3}\}$$
In other words, the solution set to this problem is $$\{\frac{\pi}{4},\frac{2\pi}{3},\frac{5\pi}{4},\frac{5\pi}{3}\}$$
