# Let $\theta\in \big(0,\large\frac{\pi}{4}\big)$ and $t_1=(\tan\theta)^{\large\tan\theta}$,$t_2=(\tan\theta)^{\large\cot\theta}$,$t_3=(\cot\theta)^{\large\tan\theta},t_4=(\cot\theta)^{\large \cot\theta}$ then

$\begin{array}{1 1}(a)\;t_1>t_2>t_3>t_4&(b)\;t_4>t_3>t_1>t_2\\(c)\;t_3>t_1>t_2>t_4&(d)\;t_2>t_3>t_1>t_4\end{array}$

Given $\theta \in \big(0,\large\frac{\pi}{4}\big)$ then $\tan\theta=1+\lambda_2$ where $\lambda_1$ and $\lambda_2$ are very small and +ve.
Then
$t_1=(1-\lambda_1)^{\large 1-\lambda_1}$
$t_2=(1-\lambda_1)^{\large 1+\lambda_1}$
$t_3=(1+\lambda_2)^{\large 1-\lambda_1}$
$t_4=(1+\lambda_2)^{\large 1+\lambda_2}$
$\Rightarrow t_4>t_3>t_1>t_2$
Hence (b) is the correct answer.
answered Oct 10, 2013