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

  1. #1
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    Lightbulb 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
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  2. #2
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    Re: Why this equation is true? Can someone explain?

    Quote Originally Posted by Fubax View Post
    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.
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  3. #3
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    Re: Why this equation is true? Can someone explain?

    This is the double angle or half angle formula for tan
    Thanks from topsquark and Fubax
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  4. #4
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    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 $\beta = \dfrac{\alpha}{2}$
    Thanks from topsquark
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    Re: Why this equation is true? Can someone explain?

    Thank you guys.
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