Question

If the range of $f\left(x\right)={\sin }^{-1}x+{\tan }^{-1}x+{\cos }^{-1}x$ is
$=[\frac{\pi }{a},\frac{b\pi }{c}]$,
Find the value of $a+b+c$

• A. $7$
• B. $8$
• C. $10$
• D. $11$

Answer 1

The correct option is D $11$

Clearly, the domain of the function is $[-1,1]$.
Also, ${\tan }^{-1}x\in [-\frac{\pi }{4},\frac{\pi }{4}]$ For $x\in [-1,1]$
Now, ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$for $x\in [-1,1]$
Thus, $f\left(x\right)={\tan }^{-1}x+\frac{\pi }{2}$, where $x\in [-1,1]$
Hence, the range is $[-\frac{\pi }{4}+\frac{\pi }{2},\frac{\pi }{4}+\frac{\pi }{2}]=[\frac{\pi }{4},\frac{3\pi }{4}]$