complex analysis

question on uniqueness theorem:

let f be an entire function s.t

f(i/n)=cos(1/n) for n a positive integer

show f(z)=cosh (z)

done this by considering S={i/n for n positive integer} noting both f and cosh agree on S and S has limit point.

the next part is

ii) find an entire function f other than f(z)=cosh(z)) that satisfies

f(in)=cos n for n a positive integer

can I just put something like

f(z)=sin(2(pi)z/i) + cos (z/i)? is there any deep justification for any function apart from cosh(z)?

can someone explain how the uniqueness theorem fails on part(ii)

thanks