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

Options

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

Answer

To solve the problem, we need to analyze the expressions \( t_1, t_2, t_3, \) and \( t_4 \) given in the question. We will evaluate these expressions for a specific value of \( \theta \) within the range \( (0, \frac{\pi}{4}) \).

### Step-by-step Solution:

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

2. **Choose a value for \( \theta \):**
Let's choose \( \theta = \frac{\pi}{6} \) (which is \( 30^\circ \)). This value lies in the interval \( (0, \frac{\pi}{4}) \).

3. **Calculate \( \tan \theta \) and \( \cot \theta \):**
\[ \tan \left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}, \quad \cot \left(\frac{\pi}{6}\right) = \sqrt{3} \]

4. **Substitute into the expressions:**
- For \( t_1 \):
\[ t_1 = \left(\frac{1}{\sqrt{3}}\right)^{\frac{1}{\sqrt{3}}} \]
- For \( t_2 \):
\[ t_2 = \left(\frac{1}{\sqrt{3}}\right)^{\sqrt{3}} = \frac{1}{3} \]
- For \( t_3 \):
\[ t_3 = \left(\sqrt{3}\right)^{\frac{1}{\sqrt{3}}} \]
- For \( t_4 \):
\[ t_4 = \left(\sqrt{3}\right)^{\sqrt{3}} \]

5. **Evaluate the expressions:**
- \( t_1 = \left(\frac{1}{\sqrt{3}}\right)^{\frac{1}{\sqrt{3}}} = 3^{-\frac{1}{6}} \)
- \( t_2 = \frac{1}{3} = 3^{-1} \)
- \( t_3 = \left(\sqrt{3}\right)^{\frac{1}{\sqrt{3}}} = 3^{\frac{1}{6}} \)
- \( t_4 = \left(\sqrt{3}\right)^{\sqrt{3}} = 3^{\frac{3}{2}} \)

6. **Compare the values:**
- \( t_4 = 3^{\frac{3}{2}} \) is the largest.
- \( t_3 = 3^{\frac{1}{6}} \) is next.
- \( t_1 = 3^{-\frac{1}{6}} \) is smaller than \( t_3 \).
- \( t_2 = 3^{-1} \) is the smallest.

7. **Conclusion:**
The order from greatest to least is:
\[ t_4 > t_3 > t_1 > t_2 \]

### Final Answer:
The correct sequence is \( t_4 > t_3 > t_1 > t_2 \).