Answer
$\dfrac{\pi-2}{2}$
Work Step by Step
Let us consider that two continuous functions $f(x)$ and $g(x)$ with $f(x)\geq g(x)$ on the interval $[a,b]$ . The area (A) of the region bounded by the graph of $f(x)$ and $g(x)$ on the interval $[a,b]$ can be calculated as: $Area(A)=\int_a^b [f(x)-g(x)] \ dx$
Thus, the area of the region is:
$A=\int_a^b [f(x)-g(x)] \ dx= \int_{0}^{\pi/4} (2-\sec^2 x)\ dx \\=[2x-\tan x]_0^{\pi/4}\\=[2(\dfrac{\pi}{4}) -\tan (\dfrac{\pi}{4}]-[(2)(0)-\tan (0)] \\=\dfrac{\pi}{2}-1\\=\dfrac{\pi-2}{2}$