Let $\theta \in \left(0, \frac{\pi}{4}\right)$ 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:
Step-by-Step Solution
Key Concept: For a fixed base, the function $a^x$ is monotonically increasing if $a > 1$ and decreasing if $0 < a < 1$.
Given $\theta \in (0, \frac{\pi}{4})$, we have $\tan \theta \in (0,1)$ and $\cot \theta \in (1, \infty)$. Define $t_1 = (\tan \theta)^{\tan \theta} \in (0,1)$, $t_2 = (\tan \theta)^{\cot \theta} \in (0,1)$, $t_3 = (\cot \theta)^{\tan \theta} \in (1, \infty)$, and $t_4 = (\cot \theta)^{\cot \theta} \in (1, \infty)$. Since numbers in $(0,1)$ decrease with increasing exponent and numbers greater than 1 increase with increasing exponent, we have $t_4 > t_3 > t_1 > t_2$.
Correct Answer: 2