Question

The range of $f(x)={\cot }^{−1}(−x)−{\tan }^{−1}x+{\sec }^{−1}x$ is

• A. $\left[−\frac{Ï€}{2},\frac{3Ï€}{2}\right]$
• B. $\left[\frac{Ï€}{2},Ï€\right)∪\left(Ï€,\frac{3Ï€}{2}\right]$
• C. $\left(\frac{−π}{2},\frac{3Ï€}{2}\right)$
• D. $\left(\frac{Ï€}{2},Ï€\right)∪\left(Ï€,\frac{3Ï€}{2}\right)$

Answer 1

Answer: (B)

We have $f(x)=π−{\cot }^{−1}x−{\tan }^{−1}x+{\sec }^{−1}x=π−\frac{π}{2}+{\sec }^{−1}x=\frac{π}{2}+{\sec }^{−1}x$
As domain of $f(x)$ is $(−∞,−1]∪[1,∞)$ $($ As ${\cot }^{−1}(−x)=π−{\cot }^{−1}x)$
$∴$ Range of $f(x)$ is $\left[\frac{π}{2},π\right)∪\left(π,\frac{3π}{2}\right]$.