What paper and question number are we looking at? (assuming it's from a past paper)
EDIT: decided to try and solve it, to see if I can help. If there's any mistakes in there, I'm sure someone will point them out.
sin^2(x) + cos^2(x) = 1
divide by sin x
1 + cot^2(x) = cosec^2(x)
rearrange for cot^2(x)
cot^2(x) = cosec^2(x) - 1
substitute into the question
4(cosec^2(x) - 1) + 12 cosec(x) + 1 = 0
Multiply out the bracket
4cosec^2(x) - 4 + 12cosec(x) + 1 = 0
Simplify
4cosec^2(x) + 12cosec(x) - 3 = 0
Doesn't factorise so put in quadratic formula
a = 4, b = 12, c = -3
cosec(x) = -(12)+-root((12)^2-4(4)(-3))/2(4) = (-3+-2root3)/2 (did on calc)
Do reciprocal to get in terms of sin(x)
sin(x) = 2/(-3+-2root3) = (6+-4root3)/3 (did on calc)
(6-4root3)/3 = -0.309, so can be this
(6+4root3)/3 = 4.309, so cant be this
sin(x) = (6-4root3)/3
Then find the solutions in the range given.