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} \leq f ( x ) \leq \frac{3 \pi^2}{4} $
$\text { We know } \frac{-\pi}{2} \leq \sin ^{-1} x \leq \frac{\pi}{2}$
$\Rightarrow \frac{4}{\pi^2} f(x) \in[-1,3] $