The simplest methods are usually the best. Try this code
N[Cos[x] - Exp[-x 27/10] /. x -> 17*^-26, 15] // InputForm or a minor variation
With[{x = 17*^-26}, N[Cos[x] - Exp[-x 27/10], 15]] // InputForm or define a function first
f = Cos[#] - Exp[-# 27/10] &; N[f[17*^-26], 15] // InputForm All of these return the result
4.589999999999999999999998802094999`15.*^-25 You can get more digits by increasing the 15 digits precision.
For example, with 34 digits precision the result returned is
4.5899999999999999999999988020950000000000000000001613`34.*^-25