Answer
Diverges.
Work Step by Step
We have:
$\int_{0}^{\pi / 2} \tan \theta d \theta =\lim\limits_{b \to (\dfrac{\pi}{2})^{-}}\int_{0}^{b} \tan \theta d \theta$
or, $=\lim\limits_{b \to (\dfrac{\pi}{2})^{-}}[-\ln |\cos \theta|]_{0}^{b}$
or, $=\lim\limits_{b \to (\dfrac{\pi}{2})^{-}}[-\ln |\cos b|+\ln 1]$
or, $=\lim\limits_{b \to (\dfrac{\pi}{2})^{-}}[-\ln |\cos b|]$
or, $=+\infty$
Thus, the limit does not exist, and so, the integral diverges.