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*}
See also:
\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.
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