So -1<=cos(stuff)<=1
So if we have an even power of cos(stuff) then
0<=cos^(2n)(stuff)<=1
Now let then power tend to infinity and you'll see the curves' peaks get steeper so that as n gets larger, most values are close to zero whereas the peaks shoot up to 1, WHEREVER THERE IS A MULTIPLE OF PI. Use geogebra to convince yourself of this (see below) 
. Now consider the 'stuff' inside the function. Now if x is a rational number, then it can be expressed as p/q where p and q are integers. After multiplying by m!, as m gets arbitrarily large, you're bound to cancel the q. Hence we are dealing with a multiple of pi and its therefore a peak and equals 1. Otherwise it equals zero.
Not a proof, just an attempt to explain intuitively. Hope that helps.
. Now consider the 'stuff' inside the function. Now if x is a rational number, then it can be expressed as p/q where p and q are integers. After multiplying by m!, as m gets arbitrarily large, you're bound to cancel the q. Hence we are dealing with a multiple of pi and its therefore a peak and equals 1. Otherwise it equals zero.
Not a proof, just an attempt to explain intuitively. Hope that helps.