Prove cos (3x) = 4cos3 (x) -3cos (x)

We need to prove cos 3x=4cos3x−3cosx

Solution

cos 3x can be written as

cos3x=cos(2x+x)—–(i)

We know the trignometric identity

cos(a+b)=cos(a)cos(b)-sin(a)sin(b)

Applying the above identity to equation (i) we get,

cos 3x =cos2xcosx−sin2xsinx

=(−1+2cos2x)cosx−2cosxsinxsinx

=−cosx+2cos3x−2sin2xcosx

=−cosx+2cos 3x−2(1−cos2x)cosx

=−3cosx+4cos3x

= 4cos3x−3cosx

Hence Proved

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