Question

If $f(x)={\left({\cos }^{-1}x\right)}^2-{\left({\sin }^{-1}x\right)}^2$ then sum of all the possible integral values of $\frac{4}{{\pi }^2}f(x)$ is

• A. 4
• B. 6
• C. 5
• D. 3

Answer 1

Answer: (C)

$f(x)={\left({\cos }^{-1}x\right)}^2-{\left({\sin }^{-1}x\right)}^2=\left({\cos }^{-1}x+{\sin }^{-1}x\right)\left({\cos }^{-1}x-{\sin }^{-1}x\right)$
$=\frac{\pi }{2}\left[\frac{\pi }{2}-2{\sin }^{-1}x\right]$
$f(x)=\frac{{\pi }^2}{4}-\pi {\sin }^{-1}x$
$\Rightarrow \frac{-{\pi }^2}{4}\le f(x)\le \frac{3{\pi }^2}{4}$
$\text{ We know }\frac{-\pi }{2}\le {\sin }^{-1}x\le \frac{\pi }{2}$
$\Rightarrow \frac{4}{{\pi }^2}f(x)\in [-1,3]$