Question

6 Let $f(x)=\sqrt{\left({\sin }^{-1}x-{\cos }^{-1}x\right)},g(x)=\sqrt{\left({\tan }^{-1}x-{\cot }^{-1}x\right)}$, then number of integer(s) in the domain of $f(x)+g(x)$ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (A)

${\sin }^{-1}x-{\cos }^{-1}x=2{\sin }^{-1}x-\frac{x}{2},f(x)=\sqrt{\left({\sin }^{-1}x-\frac{\pi }{2}\right)}$
so ${\sin }^{-1}x\ge \frac{\pi }{4}\Rightarrow \frac{1}{\sqrt{2}}\le x\le 1$ similarly
$g(x)=\sqrt{\left(2{\tan }^{-1}x-\frac{\pi }{2}\right)},{\tan }^{-1}x\ge \frac{\pi }{4}$ so $1\le x<\infty$
Domain of $(f+g)x\Rightarrow \{1\}$