Let's analyze the given expression:

$\frac{\csc^2(x) - 1}{\csc(x) - 1}$

Step 1: Use the Pythagorean Identity

We know that the Pythagorean identity for cosecant is:

$\csc^2(x) = 1 + \cot^2(x)$

However, a more useful approach is recognizing that:

$\csc^2(x) - 1 = \cot^2(x)$

Thus, we rewrite the numerator:

$\cot^2(x)$

So the expression simplifies to:

$\frac{\cot^2(x)}{\csc(x) - 1}$

Step 2: Express in Terms of Sine and Cosine

Since $\csc(x) = \frac{1}{\sin(x)}$ and $\cot(x) = \frac{\cos(x)}{\sin(x)}$, we rewrite:

$\cot^2(x) = \frac{\cos^2(x)}{\sin^2(x)}$

And

$\csc(x) - 1 = \frac{1}{\sin(x)} - 1 = \frac{1 - \sin(x)}{\sin(x)}$

Thus, the expression becomes:

$\frac{\frac{\cos^2(x)}{\sin^2(x)}}{\frac{1 - \sin(x)}{\sin(x)}}$

Dividing fractions:

$\frac{\cos^2(x)}{\sin^2(x)} \times \frac{\sin(x)}{1 - \sin(x)}$

Simplifying:

$\frac{\cos^2(x) \sin(x)}{\sin^2(x)(1 - \sin(x))}$

Canceling one $\sin(x)$:

$\frac{\cos^2(x)}{\sin(x)(1 - \sin(x))}$

This is the simplified version of the expression.

Would you like a step-by-step breakdown of any part? Here are some related questions to explore:

1. How do Pythagorean identities help in simplifying trigonometric expressions?
2. How can we simplify expressions involving cosecant and cotangent?
3. What are the steps to convert trigonometric functions to sine and cosine?
4. How do we handle division of fractions in algebraic expressions?
5. When is it useful to factor versus use an identity in trigonometry?

Tip: Always check for Pythagorean identities first when simplifying trigonometric expressions!