https://file.aiquickdraw.com/mathbot/file/1726588591267m8o3abph.jpg

Let's solve the two trigonometric problems given in the image.

### Problem 1: Prove that
\[ \tan(x) + \cot(x) = \frac{\sin(2x)}{\sin(x) \cos(x)} \]

**Solution:**

1. Recall the definitions of \(\tan(x)\) and \(\cot(x)\):
   \[ \tan(x) = \frac{\sin(x)}{\cos(x)} \]
   \[ \cot(x) = \frac{\cos(x)}{\sin(x)} \]

2. Substitute these into the left-hand side of the equation:
   \[ \tan(x) + \cot(x) = \frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)} \]

3. Find a common denominator:
   \[ \tan(x) + \cot(x) = \frac{\sin^2(x) + \cos^2(x)}{\sin(x) \cos(x)} \]

4. Use the Pythagorean identity:
   \[ \sin^2(x) + \cos^2(x) = 1 \]

5. Substitute this identity back into the equation:
   \[ \tan(x) + \cot(x) = \frac{1}{\sin(x) \cos(x)} \]

6. Recall the double-angle identity for sine:
   \[ \sin(2x) = 2 \sin(x) \cos(x) \]

7. Rewrite \(\frac{1}{\sin(x) \cos(x)}\) using the double-angle identity:
   \[ \frac{1}{\sin(x) \cos(x)} = \frac{2}{2 \sin(x) \cos(x)} = \frac{2}{\sin(2x)} \]

8. Thus,
   \[ \tan(x) + \cot(x) = \frac{\sin(2x)}{\sin(x) \cos(x)} \]

The identity is proven.

### Problem 2: Simplify the following trigonometric expression
\[ \frac{\sin(x)}{1 + \cos(x)} + \frac{1 + \cos(x)}{\sin(x)} \]

**Solution:**

1. Combine the terms over a common denominator:
   \[ \frac{\sin^2(x) + (1 + \cos(x))^2}{(1 + \cos(x)) \sin(x)} \]

2. Simplify the numerator:
   \[ \sin^2(x) + (1 + \cos(x))^2 = \sin^2(x) + 1 + 2\cos(x) + \cos^2(x) \]

3. Use the Pythagorean identity again:
   \[ \sin^2(x) + \cos^2(x) = 1 \]

4. Substitute this identity:
   \[ \sin^2(x) + 1 + 2\cos(x) + \cos^2(x) = 1 + 1 + 2\cos(x) = 2 + 2\cos(x) \]

5. Factor the numerator:
   \[ 2(1 + \cos(x)) \]

6. Substitute back into the expression:
   \[ \frac{2(1 + \cos(x))}{(1 + \cos(x)) \sin(x)} \]

7. Simplify by canceling the common factor \(1 + \cos(x)\):
   \[ \frac{2}{\sin(x)} \]

8. Recognize that \(\frac{2}{\sin(x)}\) is equivalent to \(2 \csc(x)\):

So, the simplified form is:
\[ 2 \csc(x) \]

Would you like any further details or explanations on these steps?

### Further Questions:
1. How can we derive other trigonometric identities from the basic ones?
2. What are some applications of trigonometric identities in solving real-world problems?
3. How do we solve trigonometric equations involving multiple angles?
4. What are the differences between simplifying and proving trigonometric identities?
5. How can we use trigonometric identities to solve integrals in calculus?

### Tip:
When simplifying trigonometric expressions, always look for opportunities to use fundamental identities like the Pythagorean identities, double-angle, and sum-to-product identities. These can significantly simplify the process.

Prove that tan(x) + cot(x) = sin(2x) / (sin(x) cos(x)) and simplify the expression sin(x)/(1 + cos(x)) + (1 + cos(x))/sin(x).

This problem demonstrates how to prove the trigonometric identity tan(x) + cot(x) = sin(2x) / (sin(x)cos(x)) and simplifies the expression sin(x)/(1 + cos(x)) + (1 + cos(x))/sin(x) using trigonometric identities like the Pythagorean and double-angle identities. Suitable for students in Grades 10-12.

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