I came up with it some time ago and did not understand why it's true:

Is there some specific rule for that? Maxima CAS could not tell if it's true, but it is.(Surprised)

\(\displaystyle \frac{2 \tan{\left( \frac{\alpha }{2}\right) }}{\tan{\left( \alpha \right) }}=1-{{\tan{\left( \frac{\alpha }{2}\right) }}^{2}}\)

The equation is a part of my Stereographic to Gnomonic transformation formula form Perfect Perspective shader at ReShade:

https://github.com/Fubaxiusz/fubax-shaders/blob/master/Shaders/PerfectPerspective.fx