How do you rewrite cos 3theta in terms of only cos theta?

We need to find an expression for cos3A

Solution

cos 3A can be written as

cos3A=cos(2A+A)—–(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 3A =cos2AcosA−sin2AsinA

=(−1+2cos2A)cosA−2cosAsinAsinA

=−cosA+2cos3A−2sin2AcosA

=−cosA+2cos 3A−2(1−cos2A)cosA

=−3cosA+4cos3A

Answer

Hence, cos3A= −3cosA+4cos3A

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