Question

The domain of the function $f\left(x\right)={\cos }^{-1}\left(\sec \right({\cos }^{-1}x\left)\right)+{\sin }^{-1}\left(\text{cosec}\right({\sin }^{-1}x\left)\right)$is

• A. $x\in R$
• B. $x=1,-1$
• C. $-1\le x\le 1$
• D. $x\in \varphi$

Answer 1

Answer: (B)

$x=1,-1$

$f\left(x\right)={\cos }^{-1}\left(\sec \right({\cos }^{-1}x\left)\right)+{\sin }^{-1}\left(cosec\right({\sin }^{-1}x\left)\right)$

$\Rightarrow -1\le \sec \left({\cos }^{-1}x\right)\le 1$

And $-1\le \text{cosec}\left({\sin }^{-1}x\right)\le 1$

$\Rightarrow \sec \left({\cos }^{-1}x\right)=\pm 1$

$\Rightarrow {\cos }^{-1}x=0,\pi$ and ${\sin }^{-1}x=\frac{\pi }{2},-\frac{\pi }{2}$

$\Rightarrow x=\pm 1$ and $x=\pm 1$

$\Rightarrow$ Domain is $x=\pm 1$.