Given: $\cosh x= \sec \theta$

$\therefore x - \cosh^{-1} (\sec \theta)$

Using $\cosh^{-1}(z) = \log \left ( z+ \sqrt{z^2 - 1} \right ) $

$\therefore x = \log \left ( \sec \theta + \sqrt{\sec^2 \theta - 1} \right )$

Using $\tan^2 \theta + 1 = \sec^2 \theta$

$\therefore x= \log \left ( \sec \theta + \sqrt{\tan ^2 \theta} \right )$

$\therefore \log (\sec \theta + \tan \theta )$