I would like to simplify
(a == c && c != 0) || (a == 0 && c == 0) to
a == c How can I do that?
Reduce[(a==c&&c!=0)||(a==0&&c==0),Reals] 
Not assuming a,c \[Element] Reals,
Solve[(a == c && c != 0) || (a == 0 && c == 0), c] // FullSimplify
{{c -> a}}
and
Solve[(a == c && c != 0) || (a == 0 && c == 0), a] // FullSimplify
{{a -> c}}