Answer
$$1$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /4} {{{\sec }^2}\theta } d\theta \cr
& {\text{recall that }}\int {{{\sec }^2}\theta } d\theta = \tan \theta + C \cr
& \int_0^{\pi /4} {{{\sec }^2}\theta } d\theta = \left. {\left( {\tan \theta } \right)} \right|_0^{\pi /4} \cr
& {\text{Using The Fundamental Theorem}} \cr
& = \left( {\tan \frac{\pi }{4}} \right) - \left( {\tan 0} \right) \cr
& {\text{simplify}} \cr
& = \left( 1 \right) - \left( 0 \right) \cr
& = 1 \cr} $$
