\begin{align*} \sec x + \tan x & = \tan \left( \frac{x}{2} + \frac{\pi}{4} \right) \\ & = \sqrt{\frac{1 + \sin x}{1 - \sin x}} \end{align*}

I came across this simplification while reading https://liorsinai.github.io/mathematics/2020/08/27/secant-mercator.html which talks about the integral of the secant which in turn has applications in Mercator map.