Let `t_(1)=(sin alpha)^(cos alpha), t_(2)=(sin alpha)^(sin alpha), t_(3)=(cosalpha)^(cos alpha), t_(4)=(cosalpha)^(sin alpha)`, where `alpha in (0, (pi)/(4))`, then which of the following is correct

Options

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

Answer

To solve the problem, we need to analyze the expressions given for \( t_1, t_2, t_3, \) and \( t_4 \) and compare their values based on the properties of sine and cosine functions in the interval \( \alpha \in (0, \frac{\pi}{4}) \).

### Step-by-Step Solution:

1. **Define the Expressions**:
\[ t_1 = (\sin \alpha)^{\cos \alpha}, \quad t_2 = (\sin \alpha)^{\sin \alpha}, \quad t_3 = (\cos \alpha)^{\cos \alpha}, \quad t_4 = (\cos \alpha)^{\sin \alpha} \]

2. **Analyze the Range of Sine and Cosine**:
In the interval \( \alpha \in (0, \frac{\pi}{4}) \):
- \( \sin \alpha \) is increasing and \( \sin \alpha < \frac{1}{\sqrt{2}} \)
- \( \cos \alpha \) is decreasing and \( \cos \alpha > \frac{1}{\sqrt{2}} \)

3. **Compare \( t_1 \) and \( t_2 \)**:
- Since \( \sin \alpha < \frac{1}{\sqrt{2}} \) and \( \cos \alpha < 1 \), we can analyze:
\[ t_1 = (\sin \alpha)^{\cos \alpha} \quad \text{and} \quad t_2 = (\sin \alpha)^{\sin \alpha} \]
- Since \( \sin \alpha < \cos \alpha \) in this interval, and both bases are the same, we can conclude:
\[ t_2 > t_1 \quad \text{(since the exponent for \( t_2 \) is larger)} \]

4. **Compare \( t_3 \) and \( t_4 \)**:
- Now consider:
\[ t_3 = (\cos \alpha)^{\cos \alpha} \quad \text{and} \quad t_4 = (\cos \alpha)^{\sin \alpha} \]
- Here, since \( \cos \alpha > \sin \alpha \) and both bases are the same:
\[ t_4 > t_3 \quad \text{(since the exponent for \( t_4 \) is smaller)} \]

5. **Final Comparison**:
- We have established:
\[ t_2 > t_1 \quad \text{and} \quad t_4 > t_3 \]
- Since \( \cos \alpha > \sin \alpha \) in the interval \( (0, \frac{\pi}{4}) \), we can also conclude:
\[ t_4 > t_2 \]
- Thus, we can summarize the relationships:
\[ t_4 > t_3 > t_2 > t_1 \]

6. **Conclusion**:
- The final order is:
\[ t_4 > t_3 > t_2 > t_1 \]
- Therefore, the correct option is \( t_4 > t_2 > t_1 \).