If `t_(1) = ( tan x )^(cot x ) , t_(2) = ( cot x ) ^(cot x ), t_(3) = ( tan x ) ^(tan x) , t_(4)= ( cot x )^(tan x), 0 lt x lt ( pi )/( 4)`, then `:`

Options

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

Answer

To solve the problem, we need to analyze the given expressions for \( t_1, t_2, t_3, \) and \( t_4 \) under the constraint \( 0 < x < \frac{\pi}{4} \).

### Step 1: Understanding the Trigonometric Functions
In the interval \( 0 < x < \frac{\pi}{4} \):
- \( \tan x < 1 \) (since \( \tan x \) is increasing and \( \tan \frac{\pi}{4} = 1 \))
- \( \cot x > 1 \) (since \( \cot x = \frac{1}{\tan x} \))

### Step 2: Analyzing Each Expression
1. **For \( t_1 = (\tan x)^{\cot x} \)**:
- Since \( \tan x < 1 \) and \( \cot x > 1 \), \( t_1 \) will be less than 1 because a number less than 1 raised to a power greater than 1 results in a smaller number.

2. **For \( t_2 = (\cot x)^{\cot x} \)**:
- Since \( \cot x > 1 \), raising it to its own power (which is also greater than 1) will yield a value greater than 1.

3. **For \( t_3 = (\tan x)^{\tan x} \)**:
- Similar to \( t_1 \), since \( \tan x < 1 \), \( t_3 \) will also be less than 1.

4. **For \( t_4 = (\cot x)^{\tan x} \)**:
- Here, \( \cot x > 1 \) and \( \tan x < 1 \). Since we are raising a number greater than 1 to a power less than 1, \( t_4 \) will also be greater than 1 but less than \( \cot x \).

### Step 3: Comparing the Values
Now we can summarize the relationships:
- \( t_1 < 1 \)
- \( t_3 < 1 \)
- \( t_2 > 1 \)
- \( t_4 > 1 \)

Since \( t_1 \) and \( t_3 \) are both less than 1, we can say:
\[ t_1 < t_3 \]

And since \( t_2 \) and \( t_4 \) are both greater than 1, we can say:
\[ t_2 > t_4 \]

### Step 4: Final Ordering
Putting it all together, we find:
\[ t_2 \geq t_4 > t_3 \geq t_1 \]

### Conclusion
Thus, the final order of the terms is:
\[ t_2 \geq t_4 \geq t_3 \geq t_1 \]