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.