csc(x)=1/sin(x) by definition.
Let s=sin(x), then:
1+4s(3s-4)=(1/s)(1/s-4)=1/s2-4/s.
Multiply through by s2:
s2+4s3(3s-4)=1-4s,
s2+12s4-16s3-1+4s=0,
12s4-16s3+s2+4s-1=0.
If s=1, then 12-16+1+4-1=0 so s-1 is a factor. To find another factor use synthetic division:
1 | 12 -16 1 4 -1
12 12 -4 -3 | 1
12 -4 -3 1 | 0 = 12s3-4s2-3s+1=4s2(3s-1)-(3s-1)=(2s-1)(2s+1)(3s-1).
So s=½, -½, ⅓.
Therefore sin(x)=1, ½, -½, or ⅓.