One of my students asked me "Can you write $\sin(x^2)$ in terms of $\sin(x)$"? I said I'd think about it. Having thought about it for a while, I now know that I definitely don't know the answer!
Lets relax the question to "...$\sin(x^2)$ in terms of $\sin(x)$ and $\cos(x)$".
For $x$ an integer we have $\sin(x^2)=Im[(\cos x+i\sin x)^x]$, so we can do it for fixed integers. But that isn't awfully satisfying.
Another chain of thought is that $\sin(x^2)$ isn't periodic, and so we cannot simply do something with Fourier series. However, my knowledge of Fourier series has been lost to time, and I do not know if
not periodic $\Rightarrow$ cannot be written as a sum of trig functions
is true. Sounds nice thought, doesn't it?