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identify if it is  algebra  reciprocal quotient pythagorean odd or even

To determine which rules justify each step in the proof of the trigonometric identity \((\sec^2 x - 1) \csc x = \tan x \sec x\), we will analyze the transformations to identify if they involve algebraic manipulation, reciprocal, quotient, Pythagorean identities, or properties of even and odd functions.

### Step-by-Step Justification:

1. **First Step:**  
   \[
   (\sec^2 x - 1) \csc x = \tan^2 x \csc x
   \]  
   **Rule:** **Pythagorean Identity**  
   Justification: \(\sec^2 x - 1 = \tan^2 x\).

2. **Second Step:**  
   \[
   \tan^2 x \csc x = \frac{\sin^2 x}{\cos^2 x} \csc x
   \]  
   **Rule:** **Quotient Identity**  
   Justification: \(\tan x = \frac{\sin x}{\cos x}\), so \(\tan^2 x = \frac{\sin^2 x}{\cos^2 x}\).

3. **Third Step:**  
   \[
   \frac{\sin^2 x}{\cos^2 x} \csc x = \frac{\sin^2 x}{\cos^2 x} \cdot \frac{1}{\sin x}
   \]  
   **Rule:** **Reciprocal Identity**  
   Justification: \(\csc x = \frac{1}{\sin x}\).

4. **Fourth Step:**  
   \[
   \frac{\sin^2 x}{\cos^2 x} \cdot \frac{1}{\sin x} = \frac{\sin x}{\cos^2 x}
   \]  
   **Rule:** **Algebra (Simplification)**  
   Justification: Simplify by canceling \(\sin x\) in the numerator and denominator.

5. **Fifth Step:**  
   \[
   \frac{\sin x}{\cos^2 x} = \frac{\sin x}{\cos x} \cdot \frac{1}{\cos x}
   \]  
   **Rule:** **Algebra (Multiplication)**  
   Justification: Split the fraction.

6. **Sixth Step:**  
   \[
   \frac{\sin x}{\cos x} \cdot \frac{1}{\cos x} = \tan x \sec x
   \]  
   **Rule:** **Quotient and Reciprocal Identities**  
   Justification: \(\frac{\sin x}{\cos x} = \tan x\) and \(\frac{1}{\cos x} = \sec x\).

### Summary of Rules:

1. **Pythagorean Identity**  
2. **Quotient Identity**  
3. **Reciprocal Identity**  
4. **Algebra (Simplification)**  
5. **Algebra (Multiplication)**  
6. **Quotient and Reciprocal Identities**

Would you like more details on these steps, or have any specific questions?

### Related Questions:
1. What are other examples of using the Pythagorean identity in trigonometry?
2. How do you derive the quotient identities in trigonometry?
3. Why is it useful to know the reciprocal identities?
4. What is the importance of simplifying algebraic expressions in proofs?
5. Can you explain how even and odd properties affect trigonometric identities?

**Tip:** Understanding the basic trigonometric identities and their properties can greatly simplify solving more complex proofs.

Learn how to solve the trigonometric identity (sec^2x - 1)cscx = tanx secx using key identities such as the Pythagorean Identity and Reciprocal Identity. Follow a detailed step-by-step solution suitable for high school students in Grades 10-12.

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