Complex numbers CONTRADICTION?

hey guys

1. I proved that z^-1=conj z, converting into mod-arg form and by doing de moivres theorem so this must therefore be true

1/(2+3i)=2-3i

2. yet when i substituted z for an actual complex number eg. z=2+3i

and then did the realisation the answer is different

1/(2+3i) X (2-3i)/(2-3i)=(2-3i)/13

3. and i learned of a complex number rule where

z^-1=conj z/(mod z)^2

and this rule is consistent with what got ... which was (2-3i)/13

so now the question finally is why are there 2 answers that seem right?