1. 3cot^2 (x) - 1 = 0

Question

My answer: pi/3, 2pi/3, 4pi/3, 5pi/3

2. 4cos^2 (x) - 1 = 0
My answer: pi/3, 2pi/3, 5pi/3, 4pi/3

3. 2sin (x) + csc (x) = 0
My answer: unknown lol
i got to the part: sin^2 (x) = -1/2

4. 4sin^3 (x) + 2sin^2 (x) - 2sin^2 (x) = 1

Answer 1

3. 2sin (x) + csc (x) = 0
My answer: unknown lol
i got to the part: sin^2 (x) = -1/2
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I do not know what to do either. There is no real solution.

4 looks like a typo. The second two terms are identical and cancel each other.
Are you sure it is not:
4 sin^3 x + 2 sin^2 x - 2 sin x = 1 ???
if so
let y = sin x
then
4 y^3 + 2 y^2 -2 y - 1 = 0
factor that
(2y+1) works
(2y+1)(2 y^2-1) = 0
y = sin x = -1/2
y = sin x = + ( sqrt 2) /2
y = sin x = - (sqrt 2) / 2
You can do it from there

Answer 2

Thank you!