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The image you uploaded shows a proof of a trigonometric identity where the goal is to justify each step of the proof using appropriate trigonometric rules.

Here's a breakdown of each step:

1. **Starting Expression:**
   \[
   \left( \sec^2{x} - 1 \right) \cos{x}
   \]

2. **Step 1: Pythagorean Identity:**
   \[
   \sec^2{x} - 1 = \tan^2{x}
   \]
   This step uses the **Pythagorean Identity**:
   \[
   \sec^2{x} - 1 = \tan^2{x}
   \]

3. **Step 2: Reciprocal Identity:**
   \[
   \tan^2{x} = \frac{\sin^2{x}}{\cos^2{x}}
   \]
   This step uses the **Reciprocal Identity** for tangent:
   \[
   \tan{x} = \frac{\sin{x}}{\cos{x}}
   \]

4. **Step 3: Algebraic Simplification:**
   \[
   \frac{\sin^2{x}}{\cos^2{x}} \cdot \cos{x} = \frac{\sin^2{x}}{\cos{x}}
   \]
   This step involves **Algebra** to simplify the fraction.

5. **Step 4: Quotient Identity:**
   \[
   \frac{1}{\cos{x}} = \sec{x}
   \]
   This step uses the **Quotient Identity**:
   \[
   \frac{1}{\cos{x}} = \sec{x}
   \]

6. **Final Expression:**
   \[
   \sin^2{x} \sec{x}
   \]

Would you like more details on any of these steps or have any other questions? Here are five related questions to expand on this concept:

1. How is the Pythagorean identity derived from the unit circle?
2. What are other common trigonometric identities involving secant?
3. Can this proof be approached differently using a substitution?
4. How can reciprocal identities be applied in integrals involving trigonometric functions?
5. What is the importance of quotient identities in simplifying trigonometric expressions?

**Tip:** Always simplify expressions step by step when working with trigonometric identities. Identifying which identity applies is key to solving the proof efficiently.

Complete the proof of the trigonometric identity (sec^2(x) - 1)cos(x) = sin^2(x)sec(x) by choosing the Rule that justifies each step.

Learn how to complete the proof of the trigonometric identity (sec^2(x) - 1)cos(x) = sin^2(x)sec(x) by applying key rules such as the Pythagorean, Reciprocal, and Quotient Identities. Suitable for students in Grades 10-12.

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