SOLUTION: Résoudre dans R : Arccos (2x) - Arccos(x) = π/3

Arccos (2x)  -  Arccos(x) = π/3   ===========>


arccos(2x) =  + arccos(x).


Apply cos to both sides.


cos(acrcos(2x)) =  - .    (1)


Now, part by part,


    cos(arccos(2x)) = 2x;

    cos(pi/3)       = ;

    cos(arccos(x))  = x;

     = ;

    sin(arccos(x)) = .


Substitute everything into equation (1).


    2x =  - 


    4x = x             - 

    3x = - 


Square both sides


    9x*2 = 3*(1-x^2)

    9x^2 = 3 - 3x^2

    12x^2 = 3

       x^2 =  = 

       x   = +/-  = +/- .


Now the last step is to check these two candidates.


CHECK.


Case 1.  x = .

                Then 2x = 1,  arccos(2x) = arccos(1) = 0  radians.

                arccos(x) =  = .

                 -  = - .

                The original equation IS NOT satisfied, so  x =   IS NOT the solution.



Case 2.  x = - .

                Then 2x = -1,  arccos(2x) = arccos(-1) =   radians.

                arccos(x) =  = .

                 -  = .

                The original equation IS satisfied, so  x = -   IS  the solution.


ANSWER.  The only solution to the given equation is  x = - .