solve 1 + cos 3x = cos x (1 + cos x)

Question

Cos 3x = 4cos³ x - 3cos x

Answer 1

Indeed, cos3x = cos(2x+x) = cos(2x)cosx - sin2x sinx <--- angle addition for cosine

 

                                         = (cos^2 - sin^2)cos - (2 sin cos) sin  <---- trig identities for cos2x and sin(2x);

                                                                                                            trig arguments are now just x, so it is dropped

                                                                                                             for the sake of convenience

 

                                         = cos^3 - sin^2*cos - 2 sin^2 cos 

 

                                         = cos^3 - 3 sin^2 cos 

 

                                         = cos^3 - 3 ( 1 - cos^2) cos  <---- sin^2 + cos^2 = 1 so sin^2 = 1 - cos^2

 

                                        = cos^3 - 3 (cos - cos^3) 

 

                                        = cos^3 - 3 cos + 3 cos^3

 

                                       = 4cos^3 - 3cos

which proves the identity shown in the second line

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Now to solve the equation in the first line....

 

1 + 4 cos^3 - 3cos = cos + cos^2   <--- substitutes the identity for cos 3x

 

1 + 4 cos^3 - 3cos - cos - cos^2 = 0 <---- everything to left side

 

4 cos^3 - cos^2 - 4cos + 1 = 0 <--- Combines Like Terms CLT

 

4cos^3 - cos^2 - (4cos - 1) = 0 <--- factors out -1 from last two terms

 

cos^2 (4cos - 1) - (4cos-1) = 0 <---- factors out cos^2 from first two terms

 

(cos^2 -1)(4 cos-1) = 0  <---- factors out 4cos-1  : factors by grouping

 

(cos+1)(cos-1)(4cos-1) = 0 <---- factors difference of squares pattern N^2-1 = (N+1)(N-1)

 

By zeros product property:

 cos = 1/4 OR cos = +or- 1

 

The cosine is 1/4 when inverse-cosine(0.25) = 75.52248781407....

cosine is positive in quadrants 1 and 3

 

The angles are 75.52248781407.... and 180+75.52248781407 = 255.52248781407...

 

cosine is 1 at 0 degrees and -1 at 180 degrees

 

So the solution set is { 75.52248781407...., ,   255.52248781407...,  0, 180}

 

and of course, these angles are in degrees.