Question

If the sum of maximum and minimum values of $E={\left({\sin }^{-1}x\right)}^2+2\pi {\cos }^{-1}x+{\pi }^2$ is $\frac{a{\pi }^2}{b},$ where $a$ and $b$ are coprime, then the value of $(a-b)$ is

Answer 1

Given, $E={\left({\sin }^{-1}x\right)}^2+2\pi (\frac{\pi }{2}-{\sin }^{-1}x)+{\pi }^2$
$={\left({\sin }^{-1}x\right)}^2+{\pi }^2-2\pi {\sin }^{-1}x+{\pi }^2$
$={({\sin }^{-1}x-\pi )}^2+{\pi }^2$
$\Rightarrow E_{min}={(\frac{\pi }{2}-\pi )}^2+{\pi }^2={\pi }^2+\frac{{\pi }^2}{4}=\frac{5{\pi }^2}{4}$
and $E_{max}={(-\frac{\pi }{2}-\pi )}^2+{\pi }^2={\pi }^2+\frac{9{\pi }^2}{4}=\frac{13{\pi }^2}{4}$
$\therefore E_{max}+E_{min}=\frac{18{\pi }^2}{4}=\frac{9{\pi }^2}{2}=\frac{a{\pi }^2}{b}$
$\Rightarrow$ $(a-b)=7$