Q. Let $\theta \in \left( 0,\ \frac{\pi}{4} \right)$ and $t_{1} = \left( \tan\theta \right)^{\tan\theta},\ t_{2} = \left( \tan\theta \right)^{\cot\theta},\ t_{3} = \left( \cot\theta \right)^{\tan\theta}$ and $t_{4} = \left( \cot\theta \right)^{\tan\theta}$, 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) $t_{2} > t_{3} > t_{1} > t_{4}$
Explanation
(b)
When, $\theta \in \left( 0,\ \frac{\pi}{4} \right)$
$\tan{\theta < \cot\theta}$
Since, $\tan{\theta < 1}$ and $\cot{\theta > 1}$
$\therefore\ \ \left( \tan\theta \right)^{\cot\theta} < 1$ and $\left( \cot\theta \right)^{\tan\theta} > 1$
$\therefore\ \ \ t_{4} > t_{1}$, which only holds in (b)
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.