Question

Let $\theta \in (0,\frac{\pi }{4})$ and $a={\left(\tan \theta \right)}^{\tan \theta },b={\left(\tan \theta \right)}^{\cot \theta },c={\left(\cot \theta \right)}^{\tan \theta },d={\left(\cot \theta \right)}^{\cot \theta }$, then

• A. $d>c$
• B. $b<a$
• C. $c>a$
• D. $b<a<c<d$

Answer 1

Answer: (D)

$b<a<c<d$

$\cos \theta >\sin \theta$ where $\theta \in (0,\frac{\pi }{4})$

$\Rightarrow \cot \theta >1>\tan \theta >0$ where $\theta \in (0,\frac{\pi }{4})\to \left(1\right)$

On multiplying $\left(1\right)$ with $\ln \tan \theta$ and taking antilog,

$\Rightarrow (\tan \theta {)}^{\cot \theta }<(\tan \theta {)}^{\tan \theta }$ where $\theta \in (0,\frac{\pi }{4})$ (inversion of inequality because $\ln \tan \theta <0$)

$\Rightarrow b<a$

On multiplying $\left(1\right)$ with $\ln \cot \theta$ and taking antilog,

$\Rightarrow (\cot \theta {)}^{\cot \theta }>(\cot \theta {)}^{\tan \theta }$ (Retension of inequality because $\ln \cot \theta >0$)

$\Rightarrow d>c$

On powering both side on the inequality in $\left(1\right)$ with $\tan \theta$, (note: $\tan \theta >0$)

$\Rightarrow (\cot \theta {)}^{\tan \theta }>(\tan \theta {)}^{\tan \theta }$

$\Rightarrow c>a$

ie, $b<a<c<d$

Option D is the correct answer.