Commutators of cycles in permutation groups

We prove that for n>=5, every element of the alternating group A_n is a commutator of two cycles of A_n. 
Moreover we prove that for n>= 2, a (2n+1)-cycle of the permutation group S_{2n+1} is a commutator of a 
p-cycle and a q-cycle of S_{2n+1} if and only if the following three conditions are satisfied 
(i)   n+1 < p,q, 
(ii)  2n+1 > p,q, 
(iii) p+q > 3n+1.