SOLUTION: (cos 6x + 6 cos 4x + 15 cos 2x + 10)/ (cos 5x + 5 cos 3x + 10 cos x) is equal to

(cos(6x) + 6cos(4x)  + 15cos(2x) + 10)/(cos(5x) + 5cos(3x) + 10cos(x)) 


Consider the numerator and re-group it in this way


Numerator = cos(6x) + 6*cos(4x)  + 15*cos(2x) + 10 = 

          = (cos(6x) + cos(4x)) + (5*cos(4x) + 5*cos(2x)) + (10*cos(2x) + 10)    (*)


Next, use the basic Trigonometry formula cos(a) + cos(b) = .  You will get

              cos(6x) + cos(4x) = 2*cos(5x)*cos(x), 

              cos(4x) + cos(2x) = 2*cos(3x)*cos(x), 

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


Substitute it into the formula (*). Then you can continue (*) in this way


Numerator = 2*cos(5x)*cos(x) + 5*2*cos(3x)*cos(x) + 10*2*cos(x)*cos(x) = 

          = 2*cos(x)*(cos(5x) + 5*cos(3x) + 10*cos(x))


Now notice that the long expression in the Numerator parentheses is exactly the denominator of the original formula.


Canceling common factors in the numerator and denominator, you get the final expression


     (cos(6x) + 6*cos(4x)  + 15*cos(2x) + 10) / (cos(5x) + 5*cos(3x) + 10*cos(x))  = 2*cos(x)


Answer.  (cos(6x) + 6*cos(4x)  + 15*cos(2x) + 10) / (cos(5x) + 5*cos(3x) + 10*cos(x)) = 2*cos(x).