Question

If $f\left(x\right)={\sec }^{-1}[1+{\cos }^{2}x]$ where $[.]$ denotes greatest integer function then range of $f\left(x\right)$ is :

• A. $[1,2]$
• B. $[0,2]$
• C. ${\sec }^{-1}1,{\sec }^{-1}2$
• D. $Noneofthese$

Answer 1

The correct option is C ${\sec }^{-1}1,{\sec }^{-1}2$

$-1\le \cos x\le 1$

$0\le {\cos }^{2}x\le 1$

$1\le 1+|co{s}^{2}x\le 2$

So,

$[1+{\cos }^{2}x]=\{1,2\}$

So, range of $f\left(x\right)=\left\{{\sec }^{-1}\right(1),{\sec }^{-1}(2\left)\right\}$