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Let's analyze the proof of the trigonometric identity step-by-step:

The given identity is:
\[
\tan(x)(\csc^2(x) - 1) = \cot(x)
\]

### Step-by-step Analysis:

1. **Starting expression:**
   \[
   \tan(x)(\csc^2(x) - 1)
   \]
   By definition, \(\tan(x) = \frac{\sin(x)}{\cos(x)}\).

   So,
   \[
   \frac{\sin(x)}{\cos(x)}(\csc^2(x) - 1)
   \]
   This is the application of the definition of \(\tan(x)\).

   **Rule**: Definition of \(\tan(x) = \frac{\sin(x)}{\cos(x)}\).

2. **Simplify \(\csc^2(x) - 1\):**
   Recall that \(\csc(x) = \frac{1}{\sin(x)}\), so \(\csc^2(x) = \frac{1}{\sin^2(x)}\).

   Using the Pythagorean identity: \(\csc^2(x) = 1 + \cot^2(x)\).

   Therefore,
   \[
   \csc^2(x) - 1 = \cot^2(x)
   \]
   Substituting, we get:
   \[
   \frac{\sin(x)}{\cos(x)} \cdot \cot^2(x)
   \]
   **Rule**: Pythagorean Identity: \(\csc^2(x) - 1 = \cot^2(x)\).

3. **Express \(\cot^2(x)\) in terms of \(\cos(x)\) and \(\sin(x)\):**
   Recall that \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). Therefore, \(\cot^2(x) = \frac{\cos^2(x)}{\sin^2(x)}\).

   Substituting this, we have:
   \[
   \frac{\sin(x)}{\cos(x)} \cdot \frac{\cos^2(x)}{\sin^2(x)}
   \]
   **Rule**: Definition of \(\cot(x) = \frac{\cos(x)}{\sin(x)}\).

4. **Simplify the expression:**
   Simplify the multiplication:
   \[
   \frac{\sin(x) \cdot \cos^2(x)}{\cos(x) \cdot \sin^2(x)}
   = \frac{\cos(x)}{\sin(x)}
   \]
   **Rule**: Simplification.

5. **Final result:**
   The expression \(\frac{\cos(x)}{\sin(x)}\) is equal to \(\cot(x)\).

   **Rule**: Definition of \(\cot(x) = \frac{\cos(x)}{\sin(x)}\).

### Final Answer:
The original expression \(\tan(x)(\csc^2(x) - 1) = \cot(x)\) has been verified step by step.

Would you like more details on any step, or do you have any questions?

### 5 Related Questions:
1. How can other trigonometric identities be derived using fundamental definitions?
2. What is the difference between using Pythagorean identities and quotient identities in proofs?
3. How do we verify more complex trigonometric identities involving multiple functions?
4. Why are trigonometric identities important in calculus and real-world applications?
5. Can the same identity be proven using different methods?

### Tip:
Always keep fundamental identities (like Pythagorean, reciprocal, and quotient identities) at hand when proving trigonometric equations. They make the process more straightforward.

Complete the proof of the identity by choosing the Rule that justifies each step for the given trigonometric identity: tan(x)(csc^2(x) - 1) = cot(x).

Explore the step-by-step proof of the trigonometric identity tan(x)(csc^2(x) - 1) = cot(x). This guide covers key trigonometric concepts and identities, including definitions and simplifications, suitable for high school students in Grades 10-12.

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