Answer
$${\text{The integral diverges.}}$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /2} {\sec x\tan x} dx \cr
& {\text{The integrand is not defined for }}x = \frac{\pi }{2},{\text{ then}} \cr
& \int_0^{\pi /2} {\sec x\tan x} dx = \mathop {\lim }\limits_{a \to {{\frac{\pi }{2}}^ - }} \int_0^a {\sec x\tan xdx} \cr
& {\text{Integrating}} \cr
& = \mathop {\lim }\limits_{a \to {{\frac{\pi }{2}}^ - }} \left[ {\sec x} \right]_0^a \cr
& = \mathop {\lim }\limits_{a \to {{\frac{\pi }{2}}^ - }} \left[ {\sec a - \sec 0} \right] \cr
& = \mathop {\lim }\limits_{a \to {{\frac{\pi }{2}}^ - }} \left[ {\sec a - 1} \right] \cr
& {\text{Evaluating the limit}} \cr
& = \infty \cr
& {\text{The integral diverges.}} \cr} $$