Sorry Roberto, but you've made a mistake here.
The generators of rotation around the 3 axes are the matrices
, and therefore a finite rotation through
is given by the exponentiation:
So it's not in fact the Pauli matrices
themselves we have to exponentiate (as the text would seem to indicate), but the
. (Or
for rotation about an arbitrary axis, using your notation).
You've included (without really explaining why) the factor of
, but forgotten the
. Hence the bracketed terms in your expansions of
and
are
not in fact equal to
and
as you have indicated. Recall that the power series for
sin and
cos both have alternating + and - terms; that isn't the case here!!!
My solution (which is really just the same as yours but with these problems addressed) is attached below.
