`"tan"|x| = |"tan" x|`, if

Options

A. `x in (-k pi, (2k-1)(pi)/(2)), k in Z`
B. `x in ((2k-1)(pi)/(2), k pi), k in Z`
C. `x in (-(2k+1)(pi)/(2), -k pi) cup (k pi, (2k+1)(pi)/(2)), k in Z`
D. none of these

Answer

To solve the equation \( |\tan x| = \tan |x| \), we will analyze the properties of the tangent function and the implications of the absolute values involved.

### Step 1: Understand the properties of the tangent function
The tangent function, \( \tan x \), is periodic with a period of \( \pi \) and is defined for all \( x \) except for odd multiples of \( \frac{\pi}{2} \). The absolute value \( |\tan x| \) is always non-negative.

**Hint:** Recall that the tangent function is positive in the first and third quadrants.

### Step 2: Analyze the equation
Given the equation \( |\tan x| = \tan |x| \), we need to consider the cases based on the definition of absolute values.

**Hint:** Consider the cases for \( x \) being positive and negative separately.

### Step 3: Case 1 - \( x \geq 0 \)
If \( x \geq 0 \), then \( |x| = x \). Thus, the equation simplifies to:
\[ |\tan x| = \tan x \]
This is true for all \( x \) in the first quadrant \( (0, \frac{\pi}{2}) \) and at \( x = 0 \).

**Hint:** Identify the intervals where \( \tan x \) is non-negative.

### Step 4: Case 2 - \( x < 0 \)
If \( x < 0 \), then \( |x| = -x \). The equation becomes:
\[ |\tan x| = \tan(-x) \]
Using the property \( \tan(-x) = -\tan x \), we have:
\[ |\tan x| = -\tan x \]
This implies that \( \tan x \) must be negative, which occurs in the second quadrant \( (-\frac{\pi}{2}, 0) \).

**Hint:** Determine where \( \tan x \) is negative and how that affects the equation.

### Step 5: Combine the results
From the analysis, we find that:
- For \( x \geq 0 \), \( x \) can be in the first quadrant or at \( x = 0 \).
- For \( x < 0 \), \( x \) can be in the second quadrant.

Thus, the solutions can be expressed as:
\[ x = k\pi \quad \text{(for } k \in \mathbb{Z} \text{, representing multiples of } \pi\text{)} \]
or
\[ x = 2k\pi + \frac{\pi}{2} \quad \text{(for } k \in \mathbb{Z} \text{, representing the second quadrant)} \]

### Final Answer
The complete solution set for the equation \( |\tan x| = \tan |x| \) is:
\[ x = k\pi \quad \text{or} \quad x = 2k\pi - \frac{\pi}{2} \quad (k \in \mathbb{Z}) \]