\[ \sec x + \tan x = \tan \left( \frac{x}{2} + \frac{\pi}{4} \right) = \sqrt{\frac{1 + \sin x}{1 - \sin x}} \]

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