$F(X)=F(\frac 1 X) \iff \exists G \in \mathbb{C}(X), G(X+ \frac 1 X)=F$

Question

Let $F \in \mathbb{C}(X)$ be a rational function with complex coefficients (and $\mathbb{C}(X)$ the set of such functions) i.e. $F=\frac P Q$ for $P,Q \in \mathbb{C}[X]$

I need to show that

$F(X)=F(\frac 1 X) \iff \exists G \in \mathbb{C}(X), G(X+ \frac 1 X)=F$

Showing $\Leftarrow$ is easy but I don't know how to show $\Rightarrow$.

Can someone help me please ?

Thanks in advance.