Does there exist smooth functions $f_i,g_i \in C^{\infty} (\mathbb R)$ such that $\sin (xy) = \sum\limits_{i = 1}^{n}f_i (x) g_i (y)$ for all $x,y\ $?

Question

Does there exist smooth functions $f_i,g_i \in C^{\infty} (\mathbb R)$ such that $\sin (xy) = \sum\limits_{i = 1}^{n} f_i (x) g_i (y)$ for all $x,y \in \mathbb R\ $?

I don't think it's true but couldn't able to conclude it properly. Any help in this regard would be warmly appreciated.

Thanks for your time.

Answer 1

Taking $y = 1,..., n+1$ implies that $\sin x, \sin 2x, ... , \sin((n+1)x)$ are a subset an $n$-dimensional space of functions. This will contradict that these functions are linearly independent.