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.