Verify the identity Cos 3x 4 cos3x3cosxSolutionCos 3x 4 co

Verify the identity.     Cos 3x = 4 cos^3x-3cosx

Solution

Cos 3x = 4 cos^3x-3cosx

LHS cos3x = cos(x +2x)

use trig. identity : cos(x+y) = cosx cosy -sinxsiny

sin(x +y) = sinxcosy +sinycosx

cos^2x +sin^2x = 1

Now :  cos3x = cos(x +2x)

= 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

= RHS