cos3x=4cos ^{3}x-3cosx

Formulas used: cos(a + b) = cosacosb - sinasinb sin(a + b) = sinacosb + cosasinb sin^2a + cos^2a = 1 Consider the left hand side: cos3x = cos(2x + x) = cos2xcosx - sin2xsinx = cos(x + x)cosx - sin(x + x)sinx = (cosxcosx - sinxsinx)cosx - (sinxcosx + cosxsinx)sinx = (cos^2x - sin^2x)cosx - (2sinxcosx)sinx = (cos^2x - (1 - cos^2x))cosx - 2sin^2xcosx = (cos^2x - 1 + cos^2x)cosx - 2(1 - cos^2x)cosx = (2cos^2x - 1)cosx - 2(cosx - cos^3x) = 2cos^3x - cosx - 2cosx + 2cos^3x = 4cos^3x - 3cosx = right hand side

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