Question

Range of $f\left(x\right)={\tan }^{-1}[\frac{2}{\pi }(2{\tan }^{-1}x-{\sin }^{-1}x+{\cot }^{-1}x-{\cos }^{-1}x)]$ contains

• A. Only one integer
• B. More than 2 integers
• C. Only two integers
• D. No integer

Answer 1

Answer: (A)

Only one integer

We have the range of ${\tan }^{-1}x$ is $(-\frac{\pi }{2},\frac{\pi }{2})$.

Now,

$2{\tan }^{-1}x-{\sin }^{-1}x+{\cot }^{-1}x-{\cos }^{-1}x={\tan }^{-1}x+({\tan }^{-1}x+{\cot }^{-1}x)-({\sin }^{-1}x+{\cos }^{-1}x)={\tan }^{-1}x$ [Since ${\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}$ and ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$]

As

$-\frac{\pi }{2}<{\tan }^{-1}x<\frac{\pi }{2}$

$\Rightarrow -1<\frac{2}{\pi }{\tan }^{-1}x<1$

$\Rightarrow -\frac{\pi }{4}<{\tan }^{-1}\left(\frac{2}{\pi }{\tan }^{-1}x\right)<\frac{\pi }{4}$

$\Rightarrow -0.785<f\left(x\right)<0.785$

In this range $f\left(x\right)$ has only $1$ integer value and it is $0$.