Prove the identity cos 3 4 cos3 3 cos Solutioncos 3theta4co

Prove the identity: cos 3 = 4 cos^3 3 cos

Solution

cos 3theta=4cos³theta -3cos theta

Starting with the left side

cos 3 theta=cos(2theta + theta) = cos 2theta cos theta-sin 2theta sin theta

                                               = (2cos²theta -1 ) cos theta - 2sintheta cos theta sin theta

                                              = 2cos³theta -cos theta - 2(1-cos²theta)cos theta

                                              = 2cos³theta -cos theta -2cos theta +2 cos³theta

                                              = 4cos³theta -3cos theta