To prove the identity

$\frac{1}{\sin(x)} - \frac{\sin(x)}{1 + \cos(x)} = \cot(x)$

we will use trigonometric identities and algebraic manipulations step by step, stating the rule used at each step.

Steps and Rules:

1. Rewrite $\frac{1}{\sin(x)}$ using the cosecant identity: $\frac{1}{\sin(x)} = \csc(x)$

2. Rewrite $\frac{\sin(x)}{1 + \cos(x)}$ in terms of sine and cosine: $\frac{\sin(x)}{1 + \cos(x)}$

3. Combine the terms over a common denominator: $\csc(x) - \frac{\sin(x)}{1 + \cos(x)}$

4. Express $\csc(x)$ as $\frac{1}{\sin(x)}$: $\frac{1}{\sin(x)} - \frac{\sin(x)}{1 + \cos(x)}$

5. Find a common denominator for the two fractions: $\frac{(1 + \cos(x)) - \sin^2(x)}{\sin(x)(1 + \cos(x))}$

6. Use the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ to replace $1 - \sin^2(x)$ with $\cos^2(x)$: $\frac{\cos^2(x) + \cos(x)}{\sin(x)(1 + \cos(x))}$

7. Factor $\cos(x)$ out of the numerator: $\frac{\cos(x)(\cos(x) + 1)}{\sin(x)(1 + \cos(x))}$

8. Simplify by canceling $\cos(x) + 1$ in the numerator and denominator: $\frac{\cos(x)}{\sin(x)}$

9. Recognize that $\frac{\cos(x)}{\sin(x)}$ is $\cot(x)$: $\cot(x)$

Therefore, we have proven that

$\frac{1}{\sin(x)} - \frac{\sin(x)}{1 + \cos(x)} = \cot(x)$

Steps Summary:

1. Rewrite $\frac{1}{\sin(x)}$ as $\csc(x)$.
2. Rewrite $\frac{\sin(x)}{1 + \cos(x)}$.
3. Combine over a common denominator.
4. Express $\csc(x)$ as $\frac{1}{\sin(x)}$.
5. Find a common denominator.
6. Use the Pythagorean identity.
7. Factor $\cos(x)$.
8. Cancel common factors.
9. Recognize the result as $\cot(x)$.

Questions:

1. What is the cosecant identity and how is it used in trigonometric proofs?
2. How do you find a common denominator for two trigonometric fractions?
3. What is the Pythagorean identity and how can it be applied in this context?
4. How do you factor expressions involving trigonometric functions?
5. Why is it important to simplify trigonometric expressions step by step?
6. Can you provide an example of a different trigonometric identity involving $\cot(x)$?
7. How would you prove an identity involving multiple trigonometric functions?
8. What are some common mistakes to avoid when proving trigonometric identities?

Tip:

When working with trigonometric identities, always try to rewrite everything in terms of sine and cosine, as this often simplifies the process of finding common denominators and identifying patterns.