if $x\ne 0$, is at least one of $\{x, \cos\;x\}$ transcendental over $\mathbb{Q}$?

Question

it seems at least superficially plausible that for real $x \ne 0$ then at least one of $\{x, \cos\;x\}$ is transcendental over $\mathbb{Q}$. has this assertion been proved to be true or false?

Answer 1

Prahlad Vaidyanathan's comment answers the question:

The Lindemann–Weierstrass theorem implies that if $x$ is algebraic, then $\cos(x)$ is transcendental.

So your conjecture is correct.