[tex]csc^2x+2cscx=0[/tex] Factor out the GCF which is csc x [tex]cscx(cscx+2)=0[/tex] Set each factor equal to zero. [tex]cscx=0[/tex] or [tex]cscx+2=0[/tex] Using the reciprocal sine function we can replace csc in the equations. [tex] \frac{1}{sinx} =0[/tex] or [tex] \frac{1}{sinx} +2=0[/tex] 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 [tex]sinx=- \frac{1}{2} [/tex] The sine function is negative in the 3rd and 4th quadrants. You can use the unit circle or the [tex]sin^-^1[/tex] feature on your calculator [tex]sin^-^1(1/2)[/tex] [tex]210^o or 330^o[/tex] alter radian answer which is probably what you want is [tex] \frac{7 \pi }{6} or \frac{11 \pi }{6} [/tex]
