Let (\theta \in\Bigg(0,\frac{\pi}{4}\Bigg) and t\_1=(tan \th... | Mathematics | SolveeIt

Question

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

$t_1 > t_2 > t_3 >t_4$

$t_4 > t_3 > t_1 >t_2$

$t_3 > t_1 > t_2 >t_4$

$t_2 > t_3 > t_1 >t_4$

Answer

$t_4 > t_3 > t_1 >t_2$

Explanation

Solution

The correct option is:(B): t 4>t 3>t 1>t 2.

Given: t 1=tan θ tan θ

This implies: log t 1=tan θ log(tan θ)=tan θ log(cot θ)

Which further leads to: t 1=−(cot θ tan θ)

Thus: t 2 and �1>�2 t 1>t 2

Similarly, t 4=cot θ cot θ

This implies: log⁡ log t 4=cot θ log(cot θ)=cot θ log(tan θ)

Which further leads to: t 4=−(cot θ tan θ)

Thus: t 4=− t 3 and t 4>t 3

In the range θ ∈(0,24 π ),cotθ>tanθ

⇒ t4>t3>t1>t2.
Therefore, (b) is the answer.

Mathematics Question on Trigonometric Identities

Solution