The general statement is 


    If cos(a) + cos(b) = 0, then


        EITHER  a + b =  + 

        OR      a - b =  + .


If apply it to  x  and  3x,  then


    EITHER  x + 3x =  + 

    OR      |x - 3x| =  + .


First case gives

    4x = ;  hence,  x = ,  or  x = ,  or  x = ,  or  x = .


Second case gives

   2x = ;  hence,  x = ,  or  x = .


These 6 listed values are the full set of solutions to the given equation in given interval.

Use the general formula


    cos(a) + cos(b) = .


When you apply it with  x  and  3x, you get


    cos(x) + cos(3x) = 2*cos(2x)*cos(x) = 0,


which leads you to the SAME answer.