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

Explanation for the correct option.Find the relation between the four expressions.For θ∈(0,π/4) it is seen that tanθ and also tanθand cotθ >1.So in that interv ...

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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 an...

Question

If $θ \in (0, \frac{π}{4})$ and $t_{1} = {(\tan θ)}^{\tan θ}, t_{2} = {(\tan θ)}^{c o t θ}, t_{3} = {(c o t θ)}^{\tan θ}$ and $t_{4} = {(c o t θ)}^{\tan θ},$ 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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$t_{2} > t_{3} > t_{1} > t_{4}$

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Solution

The correct option is B
$t_{4} > t_{3} > t_{1} > t_{2}$

Explanation for the correct option.
Find the relation between the four expressions.
For $θ \in (0, \frac{π}{4})$ it is seen that $\tan θ < c o t θ$ and also $\tan θ < 1$ and $c o t θ > 1$ .
So in that interval, ${(\tan θ)}^{c o t θ} < 1$ and ${(c o t θ)}^{\tan θ} > 1$
That means, $t_{4} > t_{1}$ .
From, the options given, $t_{4} > t_{1}$ can be observed only once.
Therefore, $t_{4} > t_{3} > t_{1} > t_{2}$ .
Hence, the correct option is B.

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