To solve this equation, we first notice that both terms have a factor of cos 4x. So we can factor out cos 4x:

cos^2 4x - 6 cos 4x = 0
cos 4x (cos 4x - 6) = 0

Now we solve for cos 4x:

cos 4x = 0 or cos 4x - 6 = 0

For cos 4x = 0:
4x = π/2 + nπ or 4x = 3π/2 + nπ, where n is an integer
x = π/8 + nπ/4 or x = 3π/8 + nπ/4, where n is an integer

For cos 4x - 6 = 0:
cos 4x = 6
This is not possible as the range of cos function is -1 to 1. So there are no solutions for cos 4x = 6.

Therefore, the solutions to the equation cos^2 4x - 6 cos 4x = 0 are:
x = π/8 + nπ/4 or x = 3π/8 + nπ/4, where n is an integer.