Past exam of the mathematical methods II course of the University of Oxford / 2018 / 6 / Solution

Multiply the secant function integrand by $(\sec \theta +\tan \theta )/(\sec \theta +\tan \theta )$:The numerator is the derivative of the denominator. The formula holds on any real interval avoiding zeros of $\cos \theta$; on such an interval $\sec \theta +\tan \theta$ cannot vanish because $(\sec \theta +\tan \theta )(\sec \theta -\tan \theta )=1$.