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Your last example that should yield cosh(x) had an incorrect numerator on the Prime example...you used e^x + e^y instead of e^x + e^-x.

Still the denominator could be simplified:
Sqrt[(Exp[x] + Exp[-x])^2 - (Exp[x] - Exp[-x])^2]
= 2

forget all of this ... my error, I read n! instead of n .....

Edited: 26 July 2013, 7:37 p.m. after one or more responses were posted

if n=1 the return is 1.
1!=1 and by definition 0!=1

Edited: 26 July 2013, 7:15 p.m.

Oops, you are right !
Never good to post after midnight :O

Edited: 26 July 2013, 7:27 p.m.