$F(u,v)$ is a rational function. If $\pi$ is a period of $F(\cos x,\sin x)$, then $F(u,v)=F(-u,-v)$. "Introduction to Analysis" by Teiji Takagi.

Question

I am reading "Introduction to Analysis" by Teiji Takagi.

The author wrote the following proposition without a proof.

Let $F(u,v)$ be a rational function.
If $\pi$ is a period of $F(\cos x,\sin x)$, then $F(u,v)=F(-u,-v)$ holds.

Please tell me a proof of the above proposition.

My observation:

1. Let $(u,v)\in\mathbb{R}^2-\{(0,0)\}$.
   There is a positive real number $r$ and a real number $\theta$ such that $u = r\cos\theta$ and $v=r\sin\theta$.

2. By assumption, $F(\cos x,\sin x)=F(\cos (x+\pi), \sin (x+\pi))=F(-\cos x,-\sin x)$ holds for any real number $x$ for which $F(\cos x,\sin x)$ is defined.

3. I want to show $F(r\cos x,r\sin x)=F(-r\cos x,-r\sin x)$ holds