Answer
$\pi-2$
Work Step by Step
Since $2\geq \sec^2 x$ on $ -\pi/4\leq x\leq \pi/4$
Then
\begin{align*}
A&= \int_{-\pi / 4}^{\pi / 4}\left(2-\sec ^{2} x\right) d x\\
&=\left.(2 x-\tan x)\right|_{-\pi / 4} ^{\pi / 4}\\
&=\left(\frac{\pi}{2}-1\right)-\left(-\frac{\pi}{2}+1\right)\\
&=\pi-2
\end{align*}
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