Answer
$$A = \pi - 2$$
Work Step by Step
$$\eqalign{
& {\text{From the graph we can note that the area is given by}} \cr
& A = \int_{ - \pi /4}^{\pi /4} {\left( {2 - {{\sec }^2}x} \right)} dx \cr
& {\text{Integrate}} \cr
& A = \left[ {2x - \tan x} \right]_{ - \pi /4}^{\pi /4} \cr
& {\text{Evaluate}} \cr
& A = \left[ {2\left( {\frac{\pi }{4}} \right) - \tan \left( {\frac{\pi }{4}} \right)} \right] - \left[ {2\left( { - \frac{\pi }{4}} \right) - \tan \left( { - \frac{\pi }{4}} \right)} \right] \cr
& {\text{Simplify}} \cr
& A = \left[ {\frac{\pi }{2} - 1} \right] - \left[ { - \frac{\pi }{2} + 1} \right] \cr
& A = \frac{\pi }{2} - 1 + \frac{\pi }{2} - 1 \cr
& A = \pi - 2 \cr} $$
