Question

How do you solve `4csc^2x-2=0` and find all solutions in the interval `[0,2pi)`?

Answer 1

No solutions exist for the given equation.

Explanation:

Note that the `abs(csc(theta)) >= 1` for all values of `theta`.
[See image below or note that `csc=1/sin` and `sin in [-1,+1]`

Therefore `csc^2(x) >= 1`
and `4csc^2(x) >= 4`

and finally `4csc^2(x) - 2 >= 2`

Therefore `4csc^2(x)-2 != 0` for any value of `x`