trignometry

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show that cos2x=1-tan^ 2 x /1+tan^2x=2cos^2x-1

  • trignometry -

    cos 2x
    = cos(x+x)
    = cosxcosx - sinxsinx
    = cos^2 x - sin^2 x
    = cos^2 x - (1 - cos^2 x)
    = 2 cos^2 x - 1, the last part of your equation.

    all we need it to look at now is the middle part of
    (1 - tan^2 x)/(1 + tan^2 x) , notice the necessary brackets
    = (1 - sin^2 x/cos^2 x)/(1 + sin^2 x/cos^2 x)
    = [ (cos^2 x - sin^2 x)/cos^2 x ]/[ (cos^2 x + sin^2 x)/cos^2 x ]
    = (cos^2 x - sin^2 x)/cos^2 x * cos^2 x/(cos^2 x + sin^2 x)
    = (cos^2 x - sin^2 x)/1
    = cos^2 x - sin^2 x
    as seen above

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