https://file.aiquickdraw.com/mathbot/file/1728316703586hanss1hn.jpg

We are tasked with solving the equation \(\cot(x) \sin(x) + \cot(x) = 0\).

### Step 1: Factor the expression
We can factor out \(\cot(x)\) from the equation:

\[
\cot(x) (\sin(x) + 1) = 0
\]

### Step 2: Solve for each factor
Now, we solve for each factor separately.

#### Case 1: \(\cot(x) = 0\)
The cotangent function, \(\cot(x)\), is zero when:

\[
x = \frac{\pi}{2} + n\pi \quad \text{for integers} \, n
\]

#### Case 2: \(\sin(x) + 1 = 0\)
This implies that \(\sin(x) = -1\), which happens at:

\[
x = \frac{3\pi}{2} + 2n\pi \quad \text{for integers} \, n
\]

### Step 3: Combine solutions
The solutions to the equation come from both cases:
1. \(x = \frac{\pi}{2} + n\pi\)
2. \(x = \frac{3\pi}{2} + 2n\pi\)

### Step 4: Identify the correct option
Comparing these solutions to the choices provided in the image, the correct answer matches:

\[
x = \frac{\pi}{2} + n\pi \quad \text{and} \quad x = \frac{3\pi}{2} + 2n\pi
\]

This corresponds to the second option:  
\[
\boxed{\frac{\pi}{2} + n\pi, \quad \frac{3\pi}{2} + 2n\pi}
\]

Would you like more details, or do you have any questions? Here are five related questions to expand on this:

1. What is the general form for solving trigonometric equations involving cotangent?
2. How can we interpret the periodicity of trigonometric functions in solutions?
3. Why does \(\cot(x)\) have zeros at \(\frac{\pi}{2} + n\pi\)?
4. What happens when you apply similar techniques to secant or cosecant equations?
5. How does factoring help in solving complex trigonometric equations?

**Tip:** Always check the domain of trigonometric functions, as they can introduce restrictions in certain cases.

Solve the equation cot(x)sin(x) + cot(x) = 0 and identify the correct solution from the provided options.

Learn how to solve the trigonometric equation cot(x)sin(x) + cot(x) = 0 using factoring methods and identify the correct solutions from the options. This problem involves key trigonometric concepts like cotangent and sine functions, factoring techniques, and periodic solutions.

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