Verified answer

2 cos^2(x)/sin^2(x) - 3/ sinx = 0

combine fractions

[2 cos^2(x) - 3 sin(x)] / sin^2(x) = 0

Set numerator to zero

2 cos^2(x) - 3sinx = 0

substitute cos^2(x).

2(1-sin^2(x)) - 3 sinx = 0

2 - 2sin^2(x) -3sinx = 0

Factor like a quadratic equation.

(2sinx - 1) (sinx +2) = 0

So,

sinx= 1/2 x= pi/6 and 5pi/6

sinx= -2 no solution

So all solutions between 0 and 2pi are pi/6 and 5pi/6.