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To solve the equation \(\csc^2 x - \cot x = 1\) in the interval \([0, 2\pi]\), we begin by rewriting the trigonometric identities in terms of sine and cosine:

\[
\csc x = \frac{1}{\sin x} \quad \text{and} \quad \cot x = \frac{\cos x}{\sin x}
\]

Substituting these into the equation gives:

\[
\left(\frac{1}{\sin x}\right)^2 - \frac{\cos x}{\sin x} = 1
\]

Multiply through by \(\sin^2 x\) to eliminate the denominators:

\[
1 - \cos x \sin x = \sin^2 x
\]

Now let's rearrange and use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\):

\[
1 - \cos x \sin x = 1 - \cos^2 x
\]

Simplifying this, we find:

\[
\cos x \sin x = \cos^2 x
\]

Rewriting \(\cos x \sin x\) as \(\frac{1}{2} \sin 2x\):

\[
\frac{1}{2} \sin 2x = \cos^2 x
\]

Using the identity \(\cos^2 x = \frac{1 + \cos 2x}{2}\):

\[
\frac{1}{2} \sin 2x = \frac{1 + \cos 2x}{2}
\]

Equating and simplifying:

\[
\sin 2x = 1 + \cos 2x
\]

Now, use \(\sin^2 \theta + \cos^2 \theta = 1\) for \(\theta = 2x\):

\[
\sin^2 2x = (1 + \cos 2x)^2 = 1 + 2\cos 2x + \cos^2 2x
\]

From the Pythagorean identity:

\[
1 - \cos^2 2x = 1 + 2\cos 2x + \cos^2 2x
\]

This leads to:

\[
-2\cos^2 2x - 2\cos 2x = 0
\]

Factoring out \(-2\):

\[
\cos 2x (\cos 2x + 1) = 0
\]

From here, the solutions are \(\cos 2x = 0\) or \(\cos 2x = -1\). Solving for \(x\) gives:

\[
2x = \frac{\pi}{2}, \frac{3\pi}{2} \quad \text{or} \quad 2x = \pi
\]

Dividing by 2, the solutions for \(x\) in \([0, 2\pi]\) are:

\[
x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{\pi}{2}
\]

Thus, the solutions to the equation in the given interval are:

\[
\frac{\pi}{4}, \frac{3\pi}{4}, \frac{\pi}{2}
\]

Would you like more details on any part of this solution or have any further questions?

Here are some related topics we could explore:
1. More about trigonometric identities and their uses.
2. What happens if we extend the domain beyond \(0\) to \(2\pi\).
3. Techniques for solving trigonometric equations.
4. How to graph equations involving trigonometric functions.
5. The unit circle and its relationship to trigonometric functions.

**Tip:** When solving trigonometric equations, it's often helpful to consider using fundamental identities and transformations to simplify the equation into a more manageable form.

Find all solutions of the equation csc^2(x) - cot(x) = 1 in the interval [0, 2π). Write your answer in radians in terms of π. If there is more than one solution, separate them with commas.

Find all solutions to the equation csc^2(x) - cot(x) = 1 in the interval [0, 2π). This detailed step-by-step solution explores the key trigonometric identities and transformations necessary to solve the equation. Suitable for students in Grades 10-12.

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