Answer
$$\frac{4}{\pi}$$
Work Step by Step
Using the average value equation:
$f_{\text {ave}}=\frac{1}{-a+b} \int_{a}^{b} f(x) d x$
Thus:
$f_{\text {ave}}=\frac{2}{\pi} \int_{-\pi / 4}^{\pi / 4} \sec ^{2} x d x=2 \cdot \frac{2}{\pi} =\frac{4}{\pi}$
We note:
$f(x)=\sec ^{2} x$, $b=\frac{\pi}{4}$ and $a=-\frac{\pi}{4}$
$\tan x=F(x)$