Let $\theta\in (0, \frac{\pi}{4})$ and $t_1 = (\tan\theta)^{\tan\theta}$, $t_2 = (\tan\theta)^{\cot\theta}$, $t_3=(\cot\theta)^{\tan\theta}$…

Question

Let $\theta \in (0, \frac{\pi}{4})$ 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 show that $t_4 > t_3 > t_1 > t_2$.

I don't know how to start this question. Please help. thank you:)

Answer 1

Hint:

In $0<\theta<\dfrac\pi4,$ $$\cot\theta-\tan\theta=2\cot2\theta>0$$

$\implies\cot\theta>\tan\theta$ which is $>0$

Now if $a>b>0, a^a-a^b=a^b(a^{a-b}-1)>0$

and similarly $\left(\dfrac ab\right)^a>1\implies a^a> b^a$ and so on