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 conditions under which this equality holds.

### Step 1: Understanding the Modulus Condition
The equality \( |a + b| = |a| + |b| \) holds true if both \( a \) and \( b \) are either both non-negative or both non-positive. In our case, \( a = \cot x \) and \( b = \csc x \).

### Step 2: Identify the Quadrants
1. **First Quadrant**: \( 0 < x < \frac{\pi}{2} \)
- Here, both \( \cot x \) and \( \csc x \) are positive.
2. **Second Quadrant**: \( \frac{\pi}{2} < x < \pi \)
- Here, \( \cot x \) is negative and \( \csc x \) is positive.
3. **Third Quadrant**: \( \pi < x < \frac{3\pi}{2} \)
- Here, both \( \cot x \) and \( \csc x \) are negative.
4. **Fourth Quadrant**: \( \frac{3\pi}{2} < x < 2\pi \)
- Here, \( \cot x \) is positive and \( \csc x \) is negative.

### Step 3: Analyze Each Quadrant
- **In the First Quadrant** \( (0 < x < \frac{\pi}{2}) \):
- Both \( \cot x \) and \( \csc x \) are positive.
- Thus, \( | \cot x + \csc x | = \cot x + \csc x \) and \( | \cot x | + | \csc x | = \cot x + \csc x \).
- This condition holds.

- **In the Second Quadrant** \( (\frac{\pi}{2} < x < \pi) \):
- \( \cot x < 0 \) and \( \csc x > 0 \).
- Thus, \( | \cot x + \csc x | \neq | \cot x | + | \csc x | \) since \( \cot x \) is negative and \( \csc x \) is positive.
- This condition does not hold.

- **In the Third Quadrant** \( (\pi < x < \frac{3\pi}{2}) \):
- Both \( \cot x < 0 \) and \( \csc x < 0 \).
- Thus, \( | \cot x + \csc x | = -(\cot x + \csc x) \) and \( | \cot x | + | \csc x | = -\cot x - \csc x \).
- This condition holds.

- **In the Fourth Quadrant** \( (\frac{3\pi}{2} < x < 2\pi) \):
- \( \cot x > 0 \) and \( \csc x < 0 \).
- Thus, \( | \cot x + \csc x | \neq | \cot x | + | \csc x | \) since \( \cot x \) is positive and \( \csc x \) is negative.
- This condition does not hold.

### Step 4: Conclusion
The complete set of values of \( x \) that satisfy the given equation is:
- From the first quadrant: \( x \in (0, \frac{\pi}{2}) \)
- From the third quadrant: \( x \in (\pi, \frac{3\pi}{2}) \)

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