Question

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

• 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. $t_{2}>t_{3}>t_{1}>t_{4}$

Answer 1

Answer: (B)

We have
$\theta \in\left(0, \frac{\pi}{4}\right) $
$t_{1} =(\tan \theta)^{\tan \theta} ; t_{2}=(\tan \theta)^{\cot \theta} ; t_{3}=(\cot \theta)^{\tan \theta} $
$t_{4} =(\cot \theta)^{\cot \theta}$
- For $\theta \in\left(0, \frac{\pi}{4}\right): \tan \theta \in(0,1)$ and $\cot \theta(1, \infty)$
This implies that
$t_{4}=(>1)^{>t}, t_{3}=(>1)^{<1}, t_{2}=(<1)^{>1}, t_{1}=(<1)^{<1}$
Therefore,
$t_{4}>t_{3}>t_{1}>t_{2}$