Question

If $f\left(x\right)={\tan }^{-1}x-{\cot }^{-1}x,$
$g\left(x\right)={\sec }^{-1}x-{\text{cosec}}^{-1}x,$
$h\left(x\right)={\sin }^{-1}x+{\cos }^{-1}x+{\tan }^{-1}x,$
$i\left(x\right)={\sin }^{-1}x-{\cos }^{-1}x,$
$j\left(x\right)=\sqrt{x},$
then

• A. Domain of $jof\left(x\right)+jog\left(x\right)$ is $[\sqrt{2},\infty )$
• B. Domain of $jof\left(x\right)+joi\left(x\right)$ is $[0,1]$
• C. Domain of $joi\left(x\right)+joh\left(x\right)$ is
  $[\frac{1}{\sqrt{2}},1]$
• D. Domain of $jog\left(x\right)+joh\left(x\right)$ is $\left\{1\right\}$

Answer 1

Answer: (A), (C)

Domain of $joi\left(x\right)+joh\left(x\right)$ is

$[\frac{1}{\sqrt{2}},1]$
$jof\left(x\right)=\sqrt{{\tan }^{-1}x-{\cot }^{-1}x}$
$2{\tan }^{-1}x\ge \frac{\pi }{2}$
$\Rightarrow$ Domain of $\left(jof\right(x\left)\right)=[1,\infty )\cdots \left(i\right)$

$jog\left(x\right)=\sqrt{{\sec }^{-1}x-{\text{cosec}}^{-1}x}$
$2{\sec }^{-1}x\ge \frac{\pi }{2}$
$\Rightarrow$ Domain of $\left(jog\right(x\left)\right)=(-\infty ,-1]\cup [\sqrt{2},\infty )\cdots \left(ii\right)$

$joh\left(x\right)=\sqrt{{\sin }^{-1}x+{\cos }^{-1}x+{\tan }^{-1}x}$
Domain of ${\sin }^{-1}x,{\cos }^{-1}x$ is $[-1,1]$ and
${\sin }^{-1}x+{\cos }^{-1}x+{\tan }^{-1}x\ge 0\Rightarrow {\tan }^{-1}x\ge -\frac{\pi }{2}$
$\Rightarrow$ Domain of $\left(joh\right(x\left)\right)=[-1,1]\cdots \left(iii\right)$

$joi\left(x\right)=\sqrt{{\sin }^{-1}x-{\cos }^{-1}x}$
$2{\sin }^{-1}x\ge \frac{\pi }{2}$
$\Rightarrow$ Domain of $\left(joi\right(x\left)\right)=[\frac{1}{\sqrt{2}},1]\cdots \left(iv\right)$

$\therefore$ From $\left(i\right),\left(ii\right),\left(iii\right),\left(iv\right)$
$a)$ Domain of $jof\left(x\right)+jog\left(x\right)$ is $[\sqrt{2},\infty )$
$b)$ Domain of $jof\left(x\right)+joi\left(x\right)$ is $\left\{1\right\}$
$c)$ Domain of $joi\left(x\right)+joh\left(x\right)$ is $[\frac{1}{\sqrt{2}},1]$
$d)$ Domain of $jog\left(x\right)+joh\left(x\right)$ is $\{-1\}$