Question

Solve the given equation over the interval [0, 2π): csc^2 x + 2 csc x = 0.

Answer 1

$csc^2x+2cscx=0$
Factor out the GCF which is csc x
$cscx(cscx+2)=0$
Set each factor equal to zero.
$cscx=0$  or $cscx+2=0$
Using the reciprocal sine function we can replace csc in the equations.
$\frac{1}{sinx} =0$  or $\frac{1}{sinx} +2=0$
The first equation yields 1 = 0 which is not true. However the second equation does offer 2 possible answers.
When solved for sinx you get $sinx=- \frac{1}{2}$
The sine function is negative in the 3rd and 4th quadrants.
You can use the unit circle or the $sin^-^1$ feature on your calculator  $sin^-^1(1/2)$
$210^o or 330^o$  alter radian answer which is probably what you want is $\frac{7 \pi }{6} or \frac{11 \pi }{6}$