https://file.aiquickdraw.com/mathbot/file/1726588323449vbzorqcp.jpg

Let's go through the two problems from the image one by one:

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

**Step-by-Step Proof:**

- Start with the left-hand side (LHS):
  \[
  \tan(x) + \cot(x) = \frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)}
  \]
- To simplify, combine the terms over a common denominator:
  \[
  \tan(x) + \cot(x) = \frac{\sin^2(x) + \cos^2(x)}{\sin(x)\cos(x)}
  \]
- Using the Pythagorean identity: 
  \[
  \sin^2(x) + \cos^2(x) = 1
  \]
  So the equation becomes:
  \[
  \frac{1}{\sin(x)\cos(x)}
  \]
- Next, use the identity for \(\sin(2x)\), which is:
  \[
  \sin(2x) = 2\sin(x)\cos(x)
  \]
  Therefore:
  \[
  \frac{1}{\sin(x)\cos(x)} = \frac{\sin(2x)}{\sin(x)\cos(x)}
  \]
- Hence, the left-hand side (LHS) equals the right-hand side (RHS):
  \[
  \boxed{\frac{\sin(2x)}{\sin(x)\cos(x)}}
  \]

---

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

**Step-by-Step Simplification:**

- Start by rewriting the expression:
  \[
  \frac{\sin(x)}{1 + \cos(x)} + \frac{1 + \cos(x)}{\sin(x)}
  \]
- To simplify, find a common denominator. Multiply both sides by \(\sin(x)(1 + \cos(x))\):
  \[
  \frac{\sin^2(x)}{\sin(x)(1 + \cos(x))} + \frac{(1 + \cos(x))^2}{\sin(x)(1 + \cos(x))}
  \]
  Combine them:
  \[
  \frac{\sin^2(x) + (1 + \cos(x))^2}{\sin(x)(1 + \cos(x))}
  \]
- Now expand \((1 + \cos(x))^2\):
  \[
  (1 + \cos(x))^2 = 1 + 2\cos(x) + \cos^2(x)
  \]
  So the numerator becomes:
  \[
  \sin^2(x) + 1 + 2\cos(x) + \cos^2(x)
  \]
- Using the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\):
  \[
  1 + 1 + 2\cos(x) = 2 + 2\cos(x)
  \]
- Now, the expression is:
  \[
  \frac{2(1 + \cos(x))}{\sin(x)(1 + \cos(x))}
  \]
  Cancel out the common terms:
  \[
  \frac{2}{\sin(x)}
  \]
- The simplified form is:
  \[
  \boxed{\frac{2}{\sin(x)}}
  \]

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Do you have any further questions or would you like more details on any step?

Here are 5 related questions to expand on these concepts:

1. What are the key trigonometric identities needed for simplifying expressions?
2. How can the Pythagorean identity help simplify more complex trigonometric equations?
3. Why does combining terms over a common denominator make simplifying expressions easier?
4. How does the double-angle identity \(\sin(2x)\) help in trigonometric proofs?
5. What other forms of the tangent and cotangent identities can be used in similar problems?

**Tip:** Always remember the fundamental trigonometric identities like \(\sin^2(x) + \cos^2(x) = 1\) and the double-angle identities—they're extremely useful for simplifying complex expressions.

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

Prove the identity 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 detailed step-by-step solution covers key trigonometric identities such as the Pythagorean identity and the double-angle identity. Suitable for students in Grades 10-12.

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