Match trigonometric and exponential series

Let $f(x)=a\sec x+b\tan x$ for real distinct positive constants $a,b$.

(i) Show that $f^\prime(x)=f(x)\tan x+b$.

(ii) Hence find the first three nonzero Maclaurin terms in terms of $a,b$.

(iii) These terms agree with the first three terms of $e^{\sqrt2 x}\cos x$ for small $x$. Using MF27, find the exact values of $a,b$.