We are tasked with simplifying the expression:

$\tan(x) \left( \csc(x) - \cot(x) \right)$

Let's break this down step by step:

Step 1: Recall trigonometric identities

We will use the following standard trigonometric identities: $\cot(x) = \frac{\cos(x)}{\sin(x)}, \quad \csc(x) = \frac{1}{\sin(x)}, \quad \tan(x) = \frac{\sin(x)}{\cos(x)}$

Step 2: Substitute the identities into the expression

Substitute the identities for $\cot(x)$ and $\csc(x)$ into the original expression: $\tan(x) \left( \csc(x) - \cot(x) \right) = \frac{\sin(x)}{\cos(x)} \left( \frac{1}{\sin(x)} - \frac{\cos(x)}{\sin(x)} \right)$

Step 3: Simplify the terms inside the parentheses

Inside the parentheses: $\frac{1}{\sin(x)} - \frac{\cos(x)}{\sin(x)} = \frac{1 - \cos(x)}{\sin(x)}$

Step 4: Multiply the terms

Now, we multiply $\frac{\sin(x)}{\cos(x)}$ by $\frac{1 - \cos(x)}{\sin(x)}$: $\frac{\sin(x)}{\cos(x)} \times \frac{1 - \cos(x)}{\sin(x)} = \frac{1 - \cos(x)}{\cos(x)}$

Step 5: Final simplified expression

Thus, the simplified expression is: $\frac{1 - \cos(x)}{\cos(x)}$

This is the final simplified form.

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Would you like a more detailed explanation, or do you have any questions about this? Here are some related questions you might find helpful:

1. What is the exact value of $\tan(x)$, $\csc(x)$, and $\cot(x)$ for specific angles like $30^\circ$, $45^\circ$, and $60^\circ$?
2. How can the result of trigonometric expressions like this be interpreted geometrically on the unit circle?
3. What are some alternative ways to simplify trigonometric expressions using sum and difference identities?
4. How do you solve trigonometric equations that involve $\tan(x)$, $\csc(x)$, or $\cot(x)$?
5. How do the Pythagorean identities relate to the simplification of trigonometric expressions?

Tip: Always check for opportunities to cancel terms, especially when dealing with trigonometric ratios. It can often simplify the expression significantly!