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Question

If $$t_1=(\tan x)^{\cot x}, t_2=(\cot x)^{\cot x}, t_3=(\tan x)^{\tan x}, t_4=(\cot x)^{\tan x}, 0 < x < \dfrac{\pi}{4}$$, then:


A
t1<t2<t3<t4
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B
t2>t4>t3>t1
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C
t1>t4>t3>t2
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D
t1>t2>t3>t4
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Solution

The correct option is D $$t_2 > t_4 > t_3 > t_1$$
Since $$0 < x < \dfrac{\pi}{4}$$

$$\therefore \tan x < 1$$ and $$\cot x > 1$$ .....$$(1)$$

$$\therefore$$ Choose $$\tan x=1-k_1$$ and $$\cot x=1+k_2$$, where 

$$k_1$$ and $$k_2$$ are very small $$+$$ve quantities.

$$\therefore t_3=(1-k_1)^{1-k_1}, t_1=(1-k_1)^{1+k_2}$$  ....$$(2)$$

$$t_4=(1+k_2)^{1-k_1}, t_2=(1+k_2)^{1+k_2}$$  .....$$(3)$$

$$\therefore t_4 > t_3$$ by $$(3)$$ and $$(2)$$, $$t_2 > t_4$$ by $$(2)$$ and $$(3)$$

$$\therefore t_2 > t_4 > t_3$$. Also $$t_3 > t_1$$

$$\therefore t_2 > t_4 > t_3 > t_1$$.

Mathematics

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