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

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

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