Write both angles in terms of their average.

Their average is 

 and 

Substituting in original equation,











; 

;   ;

 and  so the above is

;   

;   

;   

But since ,

,    , 

,    , 

,    , 

,    , 

Since n is an integer, n is either 0 or 1.

So the solutions are

;   

that is, x=0, and

;   

;   

So there are 3 solutions.

, , and 

 

Edwin

To simplify my writing, I will introduce the angle y =   and

will look for angles y such that  cos(6y) = cos(y)  and  0 <= y <= 90 degrees.




Since  cos(6y) = cos(y), it implies one of two possibilities:


    (1)  EITHER  6y = y + 360n  degrees

    (2)   OR     6y = -y + 360n  degrees.



From (1), I have  6y-y = 360n;  5y = 360n;   y = 0,  360/5 = 72,  (360/5)*2, . . . 

          Taking into account the restriction on the range, the only possible solutions are 0°  and  72°.



From (2), I have  6y+y = 360n;  7y = 360n;   y = 0,  360/7 = 51.43 degrees,  (360/7)*2, . . .

          Taking into account the restriction on the range, the only possible solutions are 0°  and  360/7 degrees.


It gives the ANSWER : the only solutions for x  are  0 degrees,  2*(360/5) = 144 degrees  and  2*(360/7) degrees.