Answer
The solution set to this problem is $$\{\frac{\pi}{4},\frac{3\pi}{4},\frac{7\pi}{6},\frac{11\pi}{6}\}$$
Work Step by Step
$$(\csc x+2)(\csc x-\sqrt2)=0$$ over interval $[0,2\pi)$
1) Solve the equation:
$$(\csc x+2)(\csc x-\sqrt2)=0$$
$$\csc x+2=0\hspace{1cm}\text{or}\hspace{1cm}\csc x-\sqrt2=0$$
$$\csc x=-2\hspace{1cm}\text{or}\hspace{1cm}\csc x=\sqrt2$$
2) Apply the inverse function:
- $\csc x=-2$: $\csc\lt0$ means that the angle of $x$ lies either in quadrant III or IV. In quadrant I, $\csc x=2$ refers to angle $\frac{\pi}{6}$; therefore, in quadrant III and IV, $\csc x=-2$ would refer to angle $\frac{7\pi}{6}$ and $\frac{11\pi}{6}$
- $\csc x=\sqrt2$: $\csc\gt0$ means that the angle of $x$ lies either in quadrant I or II. In quadrant I, $\csc x=\sqrt2$ refers to angle $\frac{\pi}{4}$; in quadrant II, it refers to angle $\frac{3\pi}{4}$
Therefore, $$x\in\{\frac{\pi}{4},\frac{3\pi}{4},\frac{7\pi}{6},\frac{11\pi}{6}\}$$
In other words, the solution set to this problem is $$\{\frac{\pi}{4},\frac{3\pi}{4},\frac{7\pi}{6},\frac{11\pi}{6}\}$$
