If tan ax + cot ax and |tan x| + |cot x| are periodic functions of the same fundamental then a equals

Options

A. 4
B. 2
C. 1
D. 3

Answer

To solve the problem, we need to find the value of \( a \) such that the functions \( \tan(ax) + \cot(ax) \) and \( |\tan(x)| + |\cot(x)| \) have the same fundamental period.

### Step-by-Step Solution:

1. **Identify the Period of \( \tan(ax) \) and \( \cot(ax) \)**:
- The period of \( \tan(x) \) is \( \pi \).
- The period of \( \cot(x) \) is also \( \pi \).
- Therefore, the period of \( \tan(ax) \) is given by:
\[ \text{Period of } \tan(ax) = \frac{\pi}{a} \]
- Similarly, the period of \( \cot(ax) \) is:
\[ \text{Period of } \cot(ax) = \frac{\pi}{a} \]

2. **Determine the Period of \( \tan(ax) + \cot(ax) \)**:
- Since both \( \tan(ax) \) and \( \cot(ax) \) have the same period, the period of \( \tan(ax) + \cot(ax) \) is:
\[ \text{Period of } \tan(ax) + \cot(ax) = \frac{\pi}{a} \]

3. **Identify the Period of \( |\tan(x)| + |\cot(x)| \)**:
- The period of \( |\tan(x)| \) is \( \frac{\pi}{2} \) because the absolute value function takes the negative part of the tangent function and reflects it, effectively halving the period.
- Similarly, the period of \( |\cot(x)| \) is also \( \frac{\pi}{2} \).
- Therefore, the period of \( |\tan(x)| + |\cot(x)| \) is:
\[ \text{Period of } |\tan(x)| + |\cot(x)| = \frac{\pi}{2} \]

4. **Set the Periods Equal**:
- Since it is given that the periods of both functions are equal, we can set them equal to each other:
\[ \frac{\pi}{a} = \frac{\pi}{2} \]

5. **Solve for \( a \)**:
- To solve for \( a \), we can cross-multiply:
\[ \pi \cdot 2 = \pi \cdot a \]
- This simplifies to:
\[ 2 = a \]

### Conclusion:
Thus, the value of \( a \) is:
\[ \boxed{2} \]