Answer
$$\frac{\pi }{2} - \theta $$
Work Step by Step
$$\eqalign{
& {\text{From the triangle shown bellow we have that}} \cr
& \sec \theta = \frac{{{\text{Hypotenuse}}}}{{{\text{Adjacent side}}}} \cr
& \sec \theta = x \cr
& and \cr
& {\text{From the angle }}\frac{\pi }{2} - \theta \cr
& \csc \left( {\frac{\pi }{2} - \theta } \right) = x \cr
& \cr
& {\text{Then,}} \cr
& \frac{\pi }{2} - \theta = {\csc ^{ - 1}}x \cr
& and\,\,\,\sec \theta = x \cr
& \frac{\pi }{2} - \theta = {\csc ^{ - 1}}\left( {\sec \theta } \right) \cr
& {\csc ^{ - 1}}\left( {\sec \theta } \right) = \frac{\pi }{2} - \theta \cr} $$
