Here is my signal
cos(n/2) has period 4pi and cos(pi*n/4) has period 8
Now, the question is will the signal be periodic for fundamental period 32pi ?
You can try using this:
(1) cos(a+b) = cos(a)*cos(b)+sin(a)*sin(b) (2) cos(a-b) = cos(a)*cos(b)-sin(a)*sin(b)
Then you can do (1)+(2) and get that
Then you can identify the period of cos(a+b)-cos(a-b) by using this for example http://fourier.eng.hmc.edu/e101/lectures/Fundamental_Frequency.pdf
Hope this helps