\(csc^2x+2cscx=0\)
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 \(\frac{7 \pi }{6} or \frac{11 \pi }{6}\)