Question

The solution set of inequality $(2-\frac{\pi }{2})(co{t}^{-1}x-3)+ta{n}^{-1}x(co{t}^{-1}x-3)>0$ 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)

$x\in (\cot 3,\cot 2)$

$(2-\frac{\pi }{2})({\cot }^{-1}x-3)+{\tan }^{-1}x({\cot }^{-1}x-3)>0$

$\Rightarrow ({\cot }^{-1}x-3)[2-(\frac{\pi }{2}-{\tan }^{-1}x\left)\right]>0$

$\Rightarrow ({\cot }^{-1}x-3)({\cot }^{-1}x-2)<0$ $\because {\tan }^{-1}+{\cot }^{-1}x=\frac{\pi }{2}$

$\Rightarrow {\cot }^{-1}x\in (3,2)$

$\Rightarrow x\in (\cot 3,\cot 2)$