t-statistic for correlation, $r$ \[ t = r \sqrt{\frac{n-2}{1-r^2}} \]

The reverse equation is \[ r = \frac{t}{\sqrt{t^2 + n-2}} \]

Derivation of the reverse equation: \begin{align*} t & = r \sqrt{\frac{n-2}{1-r^2}} \\ \frac{t^2}{r^2} & = \frac{n-2}{1-r^2} \\ \frac{t^2}{n-2} & = \frac{r^2}{1-r^2} \\ \frac{t^2}{t^2 + n-2} & = r^2 \\ r & = \sqrt{\frac{t^2}{t^2 + n-2}} \end{align*}

\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.