Question

3csc^2x-4=0 solve [0,2pi}

Answer 1

I'm assuming that you meant "3 (csc x)^2 - 4 = 0."  Your "csc^2x" is ambiguous.

If 3 (csc x)^2 - 4 = 0, then 3[1/sin x]^2 - 1 = 0 is equivalent.  We need to solve this for x.

First, rewrite this as 3[1/sin x]^2 = 4.
Next, divide both sides by 3:  [1/sin x]^2 = 4/3
This quadratic equation has two roots, one positive and one negative.

Supposing [1/sin x]^2 = 4/3, then  1/sin x = 2/sqrt(3).  Inverting both sides, sin x = sqrt(3)/2.   Find x by finding the inverse function:

x=arcsin sqrt(3)/2     so x = 60 degrees, or pi/3 radians.

Now find the other root or roots.