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Let's go through the steps to establish the identity:

1. **Given Identity:**
   \[
   (\tan \theta + \cot \theta) \sin \theta = \sec \theta
   \]

2. **Step 1: Express \(\tan \theta\) and \(\cot \theta\) in terms of sine and cosine.**
   - Recall that:
     \[
     \tan \theta = \frac{\sin \theta}{\cos \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta}
     \]
   - Substitute these into the expression:
     \[
     \left( \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \right) \sin \theta
     \]

3. **Step 2: Simplify the expression inside the parentheses.**
   - Combine the terms inside the parentheses over a common denominator:
     \[
     \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}
     \]
   - Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), the expression becomes:
     \[
     \frac{1}{\sin \theta \cos \theta}
     \]

4. **Step 3: Multiply by \(\sin \theta\).**
   - Now, we multiply the simplified expression by \(\sin \theta\):
     \[
     \frac{1}{\sin \theta \cos \theta} \cdot \sin \theta = \frac{1}{\cos \theta}
     \]

5. **Step 4: Final Simplification**
   - Recognize that \(\frac{1}{\cos \theta}\) is the definition of \(\sec \theta\):
     \[
     \sec \theta
     \]
   - Thus, we have shown:
     \[
     (\tan \theta + \cot \theta) \sin \theta = \sec \theta
     \]

6. **Final Answer:**
   - The fraction simplifies to \(\sec \theta\) using the **Reciprocal Identity**, as \(\sec \theta = \frac{1}{\cos \theta}\).

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Would you like more details on any of these steps, or do you have any other questions?

### Additional Related Questions
1. How do you derive the tangent and cotangent identities in terms of sine and cosine?
2. What are some examples of the Pythagorean identity used in trigonometric proofs?
3. Why is the Reciprocal Identity important in simplifying trigonometric expressions?
4. How would this identity change if \(\sin \theta\) were replaced with \(\cos \theta\)?
5. Can you explain why the expression \(\sin^2 \theta + \cos^2 \theta = 1\) holds?

### Tip:
When working with trigonometric identities, converting everything to sine and cosine can often make simplifications more straightforward.

(tan θ + cot θ) sin θ = sec θ. Establish the identity by writing the left side in terms of sine and cosine, simplifying the expression step-by-step, and identifying the identity used.

Step-by-step solution to establish the trigonometric identity (tan θ + cot θ) sin θ = sec θ. This solution covers the use of sine and cosine expressions, simplification, and the Reciprocal Identity. Suitable for high school students in Grades 10-12.

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