Find an expression for cos 3x in terms of cos x?

We need to find an expression for cos3x in terms of cos x

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

Answer

Hence, cos3x= −3cosx+4cos3x

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