Let be $E:y^2=x^3+ax^2+bx+c$ an elliptic curve, with $a,b,c\in\mathbb{Q}$, and $P=(x,y)\in E$.
I would like to show: $\forall n\in\mathbb{Z}\hspace{0.2cm}\exists \phi_n,\psi_n\in\mathbb{Q}[X]: x(nP)=\frac{\phi_n(x)}{\psi_n(x)}$
For that we can use induction. If n=2 we know that it is true due to the duplication formula.
For any $n$ we have $x(nP)=\frac{\phi_n(x,y)}{\psi_n(x,y)}$ because the addition law is given by rational functions.
Moreover we know
$x(nP)=x(-nP)=x(n(x,-y))$, so $\frac{\phi_n(x,y)}{\psi_n(x,y)} = \frac{\phi_n(x,-y)}{\psi_n(x,-y)}$
How can i finish the proof?