To find the number of solutions of the equation \( | \cot x | = \cot x + \csc x \) in the interval \( [0, 10\pi] \), we can break down the problem step by step.
### Step 1: Break down the absolute value
The equation involves the absolute value of \( \cot x \). We can consider two cases based on the definition of absolute value:
1. **Case 1**: \( \cot x \geq 0 \) (i.e., \( | \cot x | = \cot x \))
2. **Case 2**: \( \cot x < 0 \) (i.e., \( | \cot x | = -\cot x \))
### Step 2: Solve Case 1
In Case 1, we have:
\[
\cot x = \cot x + \csc x
\]
Subtracting \( \cot x \) from both sides gives:
\[
0 = \csc x
\]
Since \( \csc x = \frac{1}{\sin x} \), this implies \( \sin x \) cannot be zero (as it would make \( \csc x \) undefined). Thus, there are no solutions in this case.
### Step 3: Solve Case 2
In Case 2, we have:
\[
-\cot x = \cot x + \csc x
\]
Rearranging gives:
\[
-2 \cot x = \csc x
\]
Substituting \( \cot x = \frac{\cos x}{\sin x} \) and \( \csc x = \frac{1}{\sin x} \):
\[
-2 \frac{\cos x}{\sin x} = \frac{1}{\sin x}
\]
Multiplying through by \( \sin x \) (assuming \( \sin x \neq 0 \)):
\[
-2 \cos x = 1
\]
Thus, we have:
\[
\cos x = -\frac{1}{2}
\]
### Step 4: Find the general solutions for \( \cos x = -\frac{1}{2} \)
The solutions for \( \cos x = -\frac{1}{2} \) occur at:
\[
x = \frac{2\pi}{3} + 2k\pi \quad \text{and} \quad x = \frac{4\pi}{3} + 2k\pi \quad (k \in \mathbb{Z})
\]
### Step 5: Determine the number of solutions in the interval \( [0, 10\pi] \)
We need to find all integer values of \( k \) such that:
1. For \( x = \frac{2\pi}{3} + 2k\pi \):
\[
0 \leq \frac{2\pi}{3} + 2k\pi \leq 10\pi
\]
This simplifies to:
\[
-\frac{2\pi}{3} \leq 2k\pi \leq 10\pi - \frac{2\pi}{3}
\]
\[
-\frac{1}{3} \leq k \leq 5 - \frac{1}{3} \implies 0 \leq k \leq 5
\]
Thus, \( k = 0, 1, 2, 3, 4, 5 \) gives us **6 solutions**.
2. For \( x = \frac{4\pi}{3} + 2k\pi \):
\[
0 \leq \frac{4\pi}{3} + 2k\pi \leq 10\pi
\]
This simplifies to:
\[
-\frac{4\pi}{3} \leq 2k\pi \leq 10\pi - \frac{4\pi}{3}
\]
\[
-\frac{2}{3} \leq k \leq 5 - \frac{2}{3} \implies 0 \leq k \leq 4
\]
Thus, \( k = 0, 1, 2, 3, 4 \) gives us **5 solutions**.
### Step 6: Total solutions
Adding the solutions from both cases:
- From \( \frac{2\pi}{3} \): 6 solutions
- From \( \frac{4\pi}{3} \): 5 solutions
Thus, the total number of solutions in the interval \( [0, 10\pi] \) is:
\[
6 + 5 = 11
\]
### Final Answer
The number of solutions of the equation \( | \cot x | = \cot x + \csc x \) in the interval \( [0, 10\pi] \) is **11**.
