Let `thetain(0,pi//4)and t_(1)=(tantheta)^(tantheta), t_(2)=(tantheta)^(cottheta),t_(3)=(cot theta)^(tantheta)and t_(4)=(cot theta)^(cot theta).` Then,

Options

A. `t_(1)gtt_(2)gtt_(3)gtt_(4)`
B. `t_(4)gtt_(3)gtt_(1)gtt_(2)`
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 for \( t_1, t_2, t_3, \) and \( t_4 \) based on the given range for \( \theta \) (from \( 0 \) to \( \frac{\pi}{4} \)).

### Step 1: Define the expressions
We have:
- \( t_1 = (\tan \theta)^{\tan \theta} \)
- \( t_2 = (\tan \theta)^{\cot \theta} \)
- \( t_3 = (\cot \theta)^{\tan \theta} \)
- \( t_4 = (\cot \theta)^{\cot \theta} \)

### Step 2: Analyze the values of \( \tan \theta \) and \( \cot \theta \)
In the interval \( \theta \in [0, \frac{\pi}{4}] \):
- \( \tan \theta \) varies from \( 0 \) to \( 1 \).
- \( \cot \theta = \frac{1}{\tan \theta} \) varies from \( \infty \) (as \( \theta \) approaches \( 0 \)) to \( 1 \) (at \( \theta = \frac{\pi}{4} \)).

### Step 3: Compare \( t_1 \) and \( t_2 \)
- Since \( \tan \theta < 1 \) in this interval, \( t_1 = (\tan \theta)^{\tan \theta} < 1 \).
- \( \cot \theta > 1 \) implies \( t_2 = (\tan \theta)^{\cot \theta} \) is of the form \( (x < 1)^{y > 1} \), which is also less than \( 1 \) but greater than \( t_1 \).
- Therefore, \( t_1 < t_2 \).

### Step 4: Compare \( t_3 \) and \( t_4 \)
- \( t_3 = (\cot \theta)^{\tan \theta} \) where \( \cot \theta > 1 \) and \( \tan \theta < 1 \), thus \( t_3 > 1 \).
- \( t_4 = (\cot \theta)^{\cot \theta} \) where both base and exponent are greater than \( 1 \), thus \( t_4 > t_3 \).

### Step 5: Compare \( t_3 \) and \( t_1 \)
- We have established that \( t_1 < t_2 \) and \( t_3 > 1 \).
- Since \( t_1 < 1 \) and \( t_3 > 1 \), it follows that \( t_3 > t_1 \).

### Step 6: Combine the results
From the comparisons, we have:
1. \( t_1 < t_2 \)
2. \( t_3 > t_1 \)
3. \( t_4 > t_3 \)

Thus, we can summarize the relationships as:
\[ t_2 > t_1 \quad \text{and} \quad t_3 > t_1 \quad \text{and} \quad t_4 > t_3 \]

### Final Order
Putting it all together, we get:
\[ t_4 > t_3 > t_1 > t_2 \]

### Conclusion
The correct relationship is \( t_4 > t_3 > t_1 > t_2 \).