Answer
$c\approx^+_-.4816$
Work Step by Step
Find the values of c that are guaranteed by the mean value theorem.
$\int ^{\frac{\pi}{4}}_{\frac{-\pi}{4}} 2sec^2xdx= 2sec^2x(\frac{\pi}{4}+\frac{\pi}{4})$, Integrate
$[2tanx]^{\frac{\pi}{4}}_{\frac{-\pi}{4}}= \pi sec^2x$
$(2+2) = \pi sec^2x$
$\frac{4}{\pi}=sec^2x$
$cos^2x= \frac{\pi}{4}$
$cosx= ^+_-\sqrt {\frac{\pi}{4}}$, Use arccos
$x\approx^+_-.4816$
