Your starting equation is

    cos(2x) + cos 𝑥 + 1 = 0    (1)


Use  cos(2x) = 2cos^2(x) - 1.   Substitute it into the given equation.  You will get


    (2cos^2(x) - 1) + cos(x) + 1 = 0,   or

     2cos^2(x) + cos(x) = 0.


Factor left side


    (2cos(x) + 1)*cos(x) = 0.


So,  EITHER  cos(x) = 0, giving  x = ,  ,


     OR      2cos(x) + 1 = 0,  giving  cos(x) = - ,   x = ,  .


ANSWER.  The solutions are  ,  ,  ,  .