Answer
x = $\frac{\pi}{6}$, $\frac{5\pi}{6}$, $\frac{3\pi}{2}$
Work Step by Step
Note: csc(x) = $\frac{1}{sin(x)}$
$csc^{2}(x) - csc(x) - 2 = 0$
The expression can be factored as follows:
(csc(x) - 2) (csc(x) + 1) = 0
Since the expression is equal to 0, it can be split as follows:
csc(x) - 2 = 0 and csc(x) + 1 = 0
csc(x) - 2 = 0 can be manipulated and solved as follows:
csc(x) = 2
$\frac{1}{sin(x)} = 2$
sin(x) = $\frac{1}{2}$
x = $\frac{\pi}{6}$, $\frac{5\pi}{6}$
csc(x) + 1 = 0 can be manipulated and solved as follows:
csc(x) = -1
$\frac{1}{sin(x)} = -1$
sin(x) = -1
x = $\frac{3\pi}{2}$
The solutions are:
x = $\frac{\pi}{6}$, $\frac{5\pi}{6}$, $\frac{3\pi}{2}$
