Answer
$$\frac{4}{\pi }$$
Work Step by Step
$$\eqalign{
& {\text{Let }}f\left( t \right) = {\sec ^2}t,{\text{ on }}\underbrace {\left[ {0,\frac{\pi }{4}} \right]}_{\left[ {a,b} \right]} \cr
& {\text{The average value of }}f{\text{ on the interval }}\left[ {a,b} \right]{\text{ is}} \cr
& {f_{{\text{ave}}}} = \frac{1}{{b - a}}\int_a^b {f\left( t \right)} dt \cr
& {\text{Therefore,}} \cr
& {f_{{\text{ave}}}} = \frac{1}{{\pi /4 - 0}}\int_0^{\pi /4} {{{\sec }^2}t} dt \cr
& {f_{{\text{ave}}}} = \frac{4}{\pi }\int_0^{\pi /4} {{{\sec }^2}t} dt \cr
& {\text{Integrate}} \cr
& {f_{{\text{ave}}}} = \frac{4}{\pi }\left[ {\tan t} \right]_0^{\pi /4} \cr
& {f_{{\text{ave}}}} = \frac{4}{\pi }\left[ {\tan \left( {\frac{\pi }{4}} \right) - 0} \right] \cr
& {f_{{\text{ave}}}} = \frac{4}{\pi }\left( {1 - 0} \right) \cr
& {f_{{\text{ave}}}} = \frac{4}{\pi } \cr} $$
