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Precalculus

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Prove or disprove cos(x+y)cos(x-y)=cos squared (x) - Sin squared (x)

I dstributed the cosines and attempted to cancel out terms but I can't get the signs right. Any help on what I am missing?

  • Precalculus - ,

    LS
    = [ cosxcosy - sinxsinx] [cosxcosy + sinxsiny]
    = cos^2 x sin^2 x - sin^2 x sin^2 y
    = (cosxcosy)^2 - (sinxsiny)^2

    RS = cos^2 x - sin^2x = cos 2x

    doesn't look like they are equal

    test with values:
    let x = 60, y=45
    LS = cos(105) cos(15 = -0.25
    RS = cos^2 60 - sin^2 60 = +.25 ≠ LS
    or
    let x = 81, y= 50
    LS = cos131 cos31 = -.5623..
    RS = cos^2 81 - sin^2 81 = -.95.. ≠ LS

    not an identity

  • Precalculus - ,

    Sorry but I meant to say LS = cos^2x-sin^2y

  • Precalculus - ,

    cos(x+y)cos(x-y) = cos^2 x - sin^2 y
    (cosx cosy - sinx siny)(cosx cosy + sinx siny)
    (cosx cosy)^2 - (sinx siny)^2
    cos^2 x cos^2 y - sin^2 x sin^2 y
    cos^2 x (1-sin^2 y) - (1-cos^2 x) sin^2 y
    cos^2x - cos^2x sin^2y - sin^2y + cos^2x sin^2y
    cos^2x - sin^2y

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