More generally, simple poles (poles of order 1) occur where the Laurent series for a function has a " " term but no other negative exponents. (The order of a pole is the "highest" negative exponent in the Laurent series expansion.)
Here, the function is .
cos(z) is an analytic function for all z so its "Laurent" series is actually its Taylor series about that point and has no negative exponents. is analytic everywhere except at z= -i, so it has a Taylor series expansion about z= i and therefore so does the product . Multiplying that Taylor series by introduces a term with . Finally, just adding "4z" to the series does not change that. This function has a pole of order 1 at z= i.
For z= -i, just swap "i" and "-i".