Given $\theta \in (0,\pi /4)$ 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 

$\begin{array}{l}:\theta \in (0,\pi /4)\\ 0<\tan \theta <1\text{ and }\cot \theta >1\Rightarrow \log \cot \theta >0\end{array}$

Now,  $t_1=(\tan \theta {)}^{\tan \theta }\Rightarrow \log t_1=\tan \theta \log (\tan \theta )$

 $\begin{array}{l}\Rightarrow \log t_1=\tan \theta \log \frac{1}{\cot \theta }=\tan \theta [\log 1-\log (\cot \theta \left)\right]\\ =-\tan \theta \log (\cot \theta )\end{array}$

Similarly  $\log t_2=-\cot \theta \log (\cot \theta )$

 $\log t_3=\tan \theta \log (\cot \theta ),\log t_4=\cot \theta \log (\cot \theta )$

As $\cot \theta >\tan \theta ,$  we have 

$\begin{array}{r}\log t_4>\log t_3>\log t_1>\log t_1\\ \Rightarrow t_4>t_3>t_1>t_2.\end{array}$

Alternately $\theta \in \left(0,\frac{\pi }{4}\right)\Rightarrow 0<\tan \theta <1\Rightarrow \cot \theta >1$

Take  $\tan \theta =\frac{1}{2}\Rightarrow \cot \theta =2\text{, and we get }{t}_4>t_3>t_1>t_2$