How to prove that this two functions have simple poles:
at where k is an integer
I'm not sure I understood your definition of a 'simple pole'. Is not a pole commonly defined as the values of such that the denominator is zero? And perhaps by simple, you mean that the pole has multiplicity one?
For example, the first function would have a pole when , or equivalently . Then certainly is a pole, but since is periodic, there most certainly are other poles also.