Thread: Why this equation is true? Can someone explain?

1. Why this equation is true? Can someone explain?

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.

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

The equation is a part of my Stereographic to Gnomonic transformation formula form Perfect Perspective shader at ReShade:
https://github.com/Fubaxiusz/fubax-s...Perspective.fx

2. Re: Why this equation is true? Can someone explain?

Originally Posted by Fubax
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.

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

The equation is a part of my Stereographic to Gnomonic transformation formula form Perfect Perspective shader at ReShade:
https://github.com/Fubaxiusz/fubax-s...Perspective.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.

3. Re: Why this equation is true? Can someone explain?

This is the double angle or half angle formula for tan

4. Re: Why this equation is true? Can someone explain?

$\tan(2\beta) = \dfrac{2\tan{\beta}}{1 - \tan^2{\beta}} \implies 1-\tan^2{\beta} = \dfrac{2\tan{\beta}}{\tan(2\beta)}$

let $\beta = \dfrac{\alpha}{2}$

5. Re: Why this equation is true? Can someone explain?

Thank you guys.