If `|cot x+ cosec x|=|cot x|+ cosec x|, x in |[0,2pi],` then complete set of values of x is :

Options

A. `[0,pi]`
B. `(0, (pi)/(2)]`
C. `(0,(pi)/(2)]uu[(3pi)/(2), 2pi)`
D. `(pi, (3pi)/(2)]uu[(7pi)/(4), 2pi]`

Answer

To solve the equation \( | \cot x + \csc x | = | \cot x | + | \csc x | \) for \( x \) in the interval \( [0, 2\pi] \), we will analyze the behavior of the functions involved in different quadrants.

### Step 1: Understanding the Functions
The functions \( \cot x \) and \( \csc x \) are defined as follows:
- \( \cot x = \frac{\cos x}{\sin x} \)
- \( \csc x = \frac{1}{\sin x} \)

Both functions are undefined at \( x = 0 \) and \( x = \pi \) (where \( \sin x = 0 \)).

### Step 2: Analyzing Quadrants
We will analyze the expression in different quadrants:

1. **First Quadrant \( (0, \frac{\pi}{2}) \)**:
- Here, both \( \cot x \) and \( \csc x \) are positive.
- Thus, \( | \cot x + \csc x | = \cot x + \csc x \) and \( | \cot x | + | \csc x | = \cot x + \csc x \).
- The equation holds true.

2. **Second Quadrant \( (\frac{\pi}{2}, \pi) \)**:
- Here, \( \cot x \) is negative and \( \csc x \) is positive.
- Thus, \( | \cot x + \csc x | = -(\cot x + \csc x) \) and \( | \cot x | + | \csc x | = -\cot x + \csc x \).
- The equation does not hold true since \( -(\cot x + \csc x) \neq -\cot x + \csc x \).

3. **Third Quadrant \( (\pi, \frac{3\pi}{2}) \)**:
- Here, both \( \cot x \) and \( \csc x \) are negative.
- Thus, \( | \cot x + \csc x | = -(\cot x + \csc x) \) and \( | \cot x | + | \csc x | = -\cot x - \csc x \).
- The equation does not hold true since \( -(\cot x + \csc x) \neq -\cot x - \csc x \).

4. **Fourth Quadrant \( (\frac{3\pi}{2}, 2\pi) \)**:
- Here, \( \cot x \) is positive and \( \csc x \) is negative.
- Thus, \( | \cot x + \csc x | = \cot x - \csc x \) and \( | \cot x | + | \csc x | = \cot x - \csc x \).
- The equation holds true.

### Step 3: Valid Intervals
From the analysis:
- The valid intervals are:
- First Quadrant: \( (0, \frac{\pi}{2}) \)
- Fourth Quadrant: \( (\frac{3\pi}{2}, 2\pi) \)

### Step 4: Excluding Undefined Points
We need to exclude points where \( \csc x \) is undefined:
- At \( x = 0 \), \( \csc x \) is undefined.
- At \( x = \pi \), \( \csc x \) is undefined.
- At \( x = 2\pi \), \( \csc x \) is undefined.

### Final Solution
Thus, the complete set of values of \( x \) is:
\[ x \in \left(0, \frac{\pi}{2}\right) \cup \left(\frac{3\pi}{2}, 2\pi\right) \]