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Let's walk through the steps needed to complete this proof for the identity \( \sin(x)(\tan(x) + \cot(x)) = \sec(x) \).

1. **Distribute**:  
   \[
   \sin(x)(\tan(x) + \cot(x)) = \sin(x) \cdot \tan(x) + \sin(x) \cdot \cot(x)
   \]
   
2. **Use quotient identities**:
   - \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)
   - \( \cot(x) = \frac{\cos(x)}{\sin(x)} \)

   Substituting these into the equation:
   \[
   \sin(x) \cdot \frac{\sin(x)}{\cos(x)} + \sin(x) \cdot \frac{\cos(x)}{\sin(x)}
   \]

3. **Simplify both fractions**:
   - \( \sin(x) \cdot \frac{\sin(x)}{\cos(x)} = \frac{\sin^2(x)}{\cos(x)} \)
   - \( \sin(x) \cdot \frac{\cos(x)}{\sin(x)} = \cos(x) \)

   So now we have:
   \[
   \frac{\sin^2(x)}{\cos(x)} + \cos(x)
   \]

4. **Get a common denominator and write as a single fraction**:
   - Rewrite \( \cos(x) \) as \( \frac{\cos^2(x)}{\cos(x)} \) to have a common denominator.
   - Now we have:
     \[
     \frac{\sin^2(x) + \cos^2(x)}{\cos(x)}
     \]

5. **Use the Pythagorean identity**:
   - \( \sin^2(x) + \cos^2(x) = 1 \)
   
   So the expression becomes:
   \[
   \frac{1}{\cos(x)}
   \]

6. **Use the reciprocal identity**:
   - \( \frac{1}{\cos(x)} = \sec(x) \)

7. **Conclusion**:
   - Therefore, we have proven that:
   \[
   \sin(x)(\tan(x) + \cot(x)) = \sec(x)
   \]

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

Here are 5 related questions for further practice:
1. How would you prove the identity \( \sin(x)(\cot(x) + \tan(x)) = \csc(x) \)?
2. What is the reciprocal identity for \( \sec(x) \)?
3. How does the Pythagorean identity \( \sin^2(x) + \cos^2(x) = 1 \) simplify other trigonometric expressions?
4. Can you derive \( \tan(x) \) and \( \cot(x) \) using only sine and cosine?
5. How do common denominators help in simplifying trigonometric expressions?

**Tip**: Always look out for common trigonometric identities like \( \sin^2(x) + \cos^2(x) = 1 \) when simplifying complex expressions.

Prove the identity sin(x)(tan(x) + cot(x)) = sec(x).

Follow the step-by-step solution to prove the identity sin(x)(tan(x) + cot(x)) = sec(x). This problem uses trigonometric identities such as the Pythagorean and reciprocal identities, making it suitable for high school students in Grades 10-12.

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