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

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

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

Thank you!

To find the solutions to the given equations, you need to apply various trigonometric identities and solve for the variable. Let's go through each equation step by step:

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

Step 1: Add 1 to both sides of the equation: 3cot^2 (x) = 1.

Step 2: Divide both sides by 3: cot^2 (x) = 1/3.

Step 3: Take the square root of both sides: cot (x) = ±√(1/3).

Step 4: Find the values of x by evaluating the inverse cotangent function (cot^(-1)) of ±√(1/3).

The solutions to the equation are x = cot^(-1)(±√(1/3)), which evaluates to x = π/3, 2π/3, 4π/3, and 5π/3.

2. 4cos^2 (x) - 1 = 0

Step 1: Add 1 to both sides of the equation: 4cos^2 (x) = 1.

Step 2: Divide both sides by 4: cos^2 (x) = 1/4.

Step 3: Take the square root of both sides: cos(x) = ±√(1/4).

Step 4: Find the values of x by evaluating the inverse cosine function (cos^(-1)) of ±√(1/4).

The solutions to the equation are x = cos^(-1)(±√(1/4)), which evaluates to x = π/3, 2π/3, 4π/3, and 5π/3.

3. 2sin(x) + csc(x) = 0

Step 1: Rewrite csc(x) as 1/sin(x) to get: 2sin(x) + 1/sin(x) = 0.

Step 2: Multiply both sides by sin(x) to eliminate the fraction: 2sin^2(x) + 1 = 0.

Step 3: Rearrange the equation: 2sin^2(x) = -1.

This equation is not solvable since the square of a real number (sin(x)) cannot be negative.

Therefore, there are no solutions to this equation.

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

Step 1: Simplify the equation by combining like terms: 4sin^3(x) = 1.

Step 2: Divide both sides by 4: sin^3(x) = 1/4.

Step 3: Take the cube root of both sides: sin(x) = ∛(1/4).

Step 4: Find the values of x by evaluating the inverse sine function (sin^(-1)) of ∛(1/4).

The solution to the equation is x = sin^(-1)(∛(1/4)).