Is there no formula for $\cos(x^2)$?

I was wondering if there was a "formula" or an "identity" for $\cos(x^2)$, as there is for $\cos(2x)$. My question is closely related to this one, which was only asking for $\cos(ab)$. For instance $$\cos(x^2)= \frac{\cos^3(x)+\sin^3(x)}{2\cos^2(x)+2016}$$ could be such a formula.

More precisely, I would like to know if the function $f : x \mapsto \cos(x^2)$ belongs to $F = \Bbb R(\cos, \sin)$.

Here, the space $F$ denotes all the rational functions of the form $$x \mapsto \frac{P(\cos(x), \sin(x))}{Q(\cos(x), \sin(x))}$$ where $P, Q \in \Bbb R[X,Y]$ and $Q(\cos(x), \sin(x))≠0$ for all real numbers $x$. (Notice that $x \mapsto \cos(nx)$ belongs to $F$ (if $n≥1$). It can be proved by induction on $n$.)

My guess is no and here is why : thanks to partial fraction decomposition, every rational function has a primitive. Doing some tangent half-angle substitutions, one can see that every $g \in F$ has an explicit primitive (if I'm not mistaken).

But my function $f : x \mapsto \cos(x^2)$ above has no elementary primitive. I think this can be shown thanks to Liouville's theorem.

Does my reasoning is correct ? Do you have any easier argument (or counterargument) ?

Any comment will be appreciated !

Solution 1:

There's a simpler reason: Every rational function in $\cos x, \sin x$ has (not necessarily minimal) period $2 \pi$, but $\cos(x^2)$ is not periodic.

Solution 2:

Much easier: any rational function of $\sin$, $\cos$ will be periodic. Your function isn't.