https://file.aiquickdraw.com/mathbot/file/1726114800189fxukgahv.jpg

The expression in the image to simplify is:

\[
\frac{\sin(\theta)}{1 + \cos(\theta)} + \frac{1 + \cos(\theta)}{\sin(\theta)}
\]

Let's simplify this step-by-step.

### Step 1: Find a common denominator
The common denominator between the two terms is \( \sin(\theta)(1 + \cos(\theta)) \). We rewrite the terms with this denominator:

\[
\frac{\sin(\theta)}{1 + \cos(\theta)} + \frac{1 + \cos(\theta)}{\sin(\theta)} = \frac{\sin^2(\theta)}{\sin(\theta)(1 + \cos(\theta))} + \frac{(1 + \cos(\theta))^2}{\sin(\theta)(1 + \cos(\theta))}
\]

### Step 2: Combine the fractions
Now that we have a common denominator, combine the numerators:

\[
\frac{\sin^2(\theta) + (1 + \cos(\theta))^2}{\sin(\theta)(1 + \cos(\theta))}
\]

### Step 3: Simplify the numerator
Expand \( (1 + \cos(\theta))^2 \):

\[
(1 + \cos(\theta))^2 = 1 + 2\cos(\theta) + \cos^2(\theta)
\]

So the numerator becomes:

\[
\sin^2(\theta) + 1 + 2\cos(\theta) + \cos^2(\theta)
\]

Using the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \), we simplify the numerator:

\[
1 + 1 + 2\cos(\theta) = 2 + 2\cos(\theta)
\]

### Step 4: Factor the numerator
Factor out the 2 from the numerator:

\[
2(1 + \cos(\theta))
\]

### Step 5: Simplify the entire expression
Now, we can cancel \( 1 + \cos(\theta) \) from the numerator and denominator:

\[
\frac{2(1 + \cos(\theta))}{\sin(\theta)(1 + \cos(\theta))} = \frac{2}{\sin(\theta)}
\]

### Step 6: Recognize the result
Since \( \frac{1}{\sin(\theta)} = \csc(\theta) \), the expression simplifies to:

\[
2 \csc(\theta)
\]

Therefore, the correct answer is **2 \csc(θ)**, which matches the answer you selected.

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Do you need further details or have any questions?

Here are 5 related questions:
1. What is the Pythagorean identity and how is it used in trigonometric simplifications?
2. How do you find a common denominator for trigonometric fractions?
3. Why is it important to factor expressions in trigonometry?
4. How does recognizing trigonometric identities help in solving complex problems?
5. Can you apply a similar process to simplify different trigonometric expressions?

**Tip:** Always look for common trigonometric identities like Pythagorean or reciprocal identities when simplifying complex expressions.

Which of the following is equivalent to sin(θ) / (1 + cos(θ)) + (1 + cos(θ)) / sin(θ)?

Learn how to simplify the trigonometric expression sin(θ) / (1 + cos(θ)) + (1 + cos(θ)) / sin(θ). This problem involves using the Pythagorean and reciprocal identities, suitable for high school students studying trigonometry.

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