If -0- -4 and t1-tan-tan- t2-tan- t3-tan- and t4- then

The correct option is C t_4 gt; t_3 gt; t_1 gt; t_2θ∈(0, π4) ⇒tanθ lt; 1 and θ gt; 1. Let tanθ = 1 x and θ = 1 + y. where x, y gt; 0 and are ...

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Byju's Answer

Standard XII

Mathematics

Tangent, Cotangent, Secant, Cosecant in Terms of Sine and Cosine

If θ∈0, π/4 a...

Question

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

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

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

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

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t_{2} > t_{3} > t_{1} > t_{2}

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Solution

The correct option is B $t_{4} > t_{3} > t_{1} > t_{2}$
$θ \in (0, \frac{π}{4}) \Rightarrow tan θ < 1$ 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 $t_{3} > t_{1} .$
Thus, $t_{4} > t_{3} > t_{1} > t_{2} .$

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If θ∈(0,π4) and t1=(tanθ)tanθ, t2=(tanθ)cotθ, t3=(cotθ)tanθ and t4=(cotθ)cotθ then

The correct option is B t4>t3>t1>t2θ∈(0,π4)⇒tanθ<1 and cotθ>1. Let tanθ=1−x and cotθ=1+y. where x,y>0 and are very small, then t1=(1−x)1−x,t2=(1−x)1+y,t3=(1+y)1−x,t4=(1+y)1+y Clearly, t4>t3 and t1>t2. Also t3>t1. Thus, t4>t3>t1>t2.

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