Question

The solution set of inequality $\left({\cot }^{-1}x\right)\left({\tan }^{-1}x\right)+\left(2-\frac{\pi }{2}\right){\cot }^{-1}x-3{\tan }^{-1}x-3\left(2-\frac{\pi }{2}\right)>0$, is

• A. $x\in (\tan 2,\tan 3)$
• B. $x\in (\cot 3,\cot 2)$
• C. $x\in (-\infty ,\tan 2)\cup (\tan 3,\infty )$
• D. $x\in (-\infty ,\cot 3)\cup (\cot 2,\infty )$

Answer 1

Answer: (B)

Given $\left({\cot }^{-1}x\right)\left({\tan }^{-1}x\right)+\left(2-\frac{\pi }{2}\right){\cot }^{-1}x-3{\tan }^{-1}x-3\left(2-\frac{\pi }{2}\right)>0$
$\Rightarrow {\cot }^{-1}x\left({\tan }^{-1}x+2-\frac{\pi }{2}\right)-3\left({\tan }^{-1}x+2-\frac{\pi }{2}\right)>0$
$($ As ${\tan }^{-1}x-\frac{\pi }{2}=-{\cot }^{-1}x)$
$\Rightarrow \left({\cot }^{-1}x-3\right)\left(2-{\cot }^{-1}x\right)>0$ $\Rightarrow \left({\cot }^{-1}x-3\right)\left({\cot }^{-1}x-2\right)<0$
$\Rightarrow 2<{\cot }^{-1}x<3\Rightarrow \cot 3<x<\cot 2$ (As ${\cot }^{-1}x$ is a decreasing function.)
Hence $x\in (\cot 3,\cot 2)$