Let ( 0 , 4 ) and t 1 = ( tan ) tan , t 2 = ( tan ) cot , t 3 = ( cot ) tan , t 4 = ( cot ) cot , then [2006]

Question

Let $θ \in (0, \frac{π}{4})$ and $t_{1} = {(\tan θ)}^{\tan θ}, t_{2} = {(\tan θ)}^{\cot θ}, t_{3} = {(\cot θ)}^{\tan θ}, t_{4} = {(\cot θ)}^{\cot θ},$ then [2006]

Smart Achievers · Accepted Answer

(2)

$Given: θ \in (0, \frac{π}{4}) \Rightarrow \tan θ < 1 and \cot θ > 1$

$Let \tan θ = 1 - x and \cot θ = 1 + y, where x, y > 0 and are very small, then$

$∴$ $t_{1} = {(1 - x)}^{1 - x}, t_{2} = {(1 - x)}^{1 + y}, t_{3} = {(1 + y)}^{1 - x}, t_{4} = {(1 + y)}^{1 + y}$

$Clearly, t_{4} > t_{3} and t_{1} > t_{2} also, t_{3} > t_{1}$

$∴$ $t_{4} > t_{3} > t_{1} > t_{2}$

Solution

Q.
Let $θ \in (0, \frac{π}{4})$ and $t_{1} = {(\tan θ)}^{\tan θ}, t_{2} = {(\tan θ)}^{\cot θ}, t_{3} = {(\cot θ)}^{\tan θ}, t_{4} = {(\cot θ)}^{\cot θ},$ then [2006]

1 $t_{1} > t_{2} > t_{3} > t_{4}$

2 $t_{4} > t_{3} > t_{1} > t_{2}$

3 $t_{3} > t_{1} > t_{2} > t_{4}$

4 $t_{2} > t_{3} > t_{1} > t_{4}$

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Solution

Q. Let θ∈(0,π4) and t1=(tanθ)tanθ, t2=(tanθ)cotθ, t3=(cotθ)tanθ, t4=(cotθ)cotθ, then [2006]

1 t1>t2>t3>t4

2 t4>t3>t1>t2

3 t3>t1>t2>t4

4 t2>t3>t1>t4