Verify the identity...

Mathematics
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cos3x= cos^3x - 3sin^2xcosx 

Feb 1st, 2015

cos3x = cos(2x +x)

           = cos2x cosx - sin2x sinx  ,  cos(x +y) = cosx cosy - sinx siny

          = (2cos^2x - 1) cosx - (2sinx cosx) sinx  ,  using   cos2x = 2cos^2x - 1 & sin2x = 2sinx cosx

         = 2cos^3x - cosx - 2sin^2x cosx

       =      2cos^3x - cosx - 2(1 - cos^2x) cosx  ,  sin^2x = 1 - cos^2x

      = 4cos^3x - 3cosx   ,  simplify

      = cos^3x + 3cos^3x - 3cosx

     = cos^3x - 3cosx( 1 - cos^2x)  , take 3cosx common

      = cos^3x - 3cosx sin^2x  ,sin^2x = 1 - cos^2x

Feb 1st, 2015

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