Let `theta in (0,pi/4) and t_1=(tan theta)^(tan theta), t_2=(tan theta)6(cot theta), t_3=(cot theta)^(tan theta) and t_4=(cot theta)^(cot theta),` then

Options

A. `t_(1)gtt_(2)gtt_(3)gtt_(4)`
B. `t_(4)gtt_(3)gtt_(2)gtt_(1)`
C. `t_(3)gtt_(1)gtt_(2)gtt_(4)`
D. `t_(2)gtt_(3)gtt_(1)gtt_(4)`

Answer

To solve the problem, we need to analyze the expressions given for \( t_1, t_2, t_3, \) and \( t_4 \) based on the values of \( \tan \theta \) and \( \cot \theta \) for \( \theta \in (0, \frac{\pi}{4}) \).

### Step-by-Step Solution:

1. **Define the expressions:**
- \( t_1 = (\tan \theta)^{\tan \theta} \)
- \( t_2 = (\tan \theta)^{\cot \theta} \)
- \( t_3 = (\cot \theta)^{\tan \theta} \)
- \( t_4 = (\cot \theta)^{\cot \theta} \)

2. **Understand the behavior of \( \tan \theta \) and \( \cot \theta \):**
- For \( \theta \in (0, \frac{\pi}{4}) \):
- \( \tan \theta \) is in the interval \( (0, 1) \)
- \( \cot \theta = \frac{1}{\tan \theta} \) is in the interval \( (1, \infty) \)

3. **Analyze \( t_1 \):**
- Since \( \tan \theta < 1 \), \( t_1 = (\tan \theta)^{\tan \theta} \) will be a positive number less than 1.

4. **Analyze \( t_2 \):**
- \( t_2 = (\tan \theta)^{\cot \theta} \)
- Since \( \tan \theta < 1 \) and \( \cot \theta > 1 \), \( t_2 \) will also be a positive number less than 1, but it will be smaller than \( t_1 \) because raising a number less than 1 to a power greater than 1 results in a smaller number.

5. **Analyze \( t_3 \):**
- \( t_3 = (\cot \theta)^{\tan \theta} \)
- Here, \( \cot \theta > 1 \) and \( \tan \theta < 1 \), so \( t_3 \) will be a positive number greater than 1 but less than \( \cot \theta \).

6. **Analyze \( t_4 \):**
- \( t_4 = (\cot \theta)^{\cot \theta} \)
- Since \( \cot \theta > 1 \), \( t_4 \) will be a positive number greater than 1 and significantly larger than \( t_3 \).

7. **Comparison of the values:**
- From the analysis:
- \( t_4 > t_3 > t_1 > t_2 \)
- Therefore, the order of the values is:
- \( t_4 > t_3 > t_1 > t_2 \)

### Conclusion:
The greatest value is \( t_4 \), followed by \( t_3 \), then \( t_1 \), and the smallest is \( t_2 \).