Why this equation is true? Can someone explain?

Aug 2019
2
0
Europe
Can someone explain why this equation is true?
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
 

Plato

MHF Helper
Aug 2006
22,472
8,642
Can someone explain why this equation is true?
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
SEE HERE scroll down the page to see it is true.
Frankly I have not been able to find the path way to doing this yet.
 
Jun 2013
1,110
590
Lebanon
This is the double angle or half angle formula for tan
 
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Dec 2014
130
101
USA
$\tan(2\beta) = \dfrac{2\tan{\beta}}{1 - \tan^2{\beta}} \implies 1-\tan^2{\beta} = \dfrac{2\tan{\beta}}{\tan(2\beta)}$

let $\beta = \dfrac{\alpha}{2}$
 
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