I will use (a,b) to represent the gcd of a and b and write Fn for F(n).
The hint given is a special case of the non trivial result (a,b) = (a,b+ax) for any x.
The base case n=1 is trivial. Suppose the result holds for n. Now (Fn+1,Fn) = (Fn+Fn-1, Fn) by definition of Fn. Since (a,b) = (b,a) so (Fn+1,Fn) = (Fn+Fn-1, Fn) = (Fn,Fn + Fn-1). Also by putting a =Fn, b = Fn-1 and x = 1 in (a,b) = (a,b+ax) we have (Fn,Fn + Fn-1) = (Fn,Fn-1). By the induction hypothesis (Fn,Fn-1) is 1. So (Fn+1,Fn) =1.