Let $f(x)$ be a real transcendental function with algebraic coefficients. So $f(x)$ and $x$ are algebraically independent. Let $\alpha$ be a transcendental number, are the numbers $$\alpha+f(\alpha),\ \alpha f(\alpha),\ \alpha/f(\alpha)$$ transcendental?
It is clear these numbers are transcendental if $f(\alpha)$ is algebraic, but what if $f(\alpha)$ is transcendental? It seems in this case these numbers are also transcendental because $f(x)$ and $x$ are algebraically independent, but I'm not able to show this. I'm sorry maybe the description above is not satisfactoty. Here is a specific version: If $\alpha$ is transcendental, then are $$\alpha+\cos(\alpha), \ \alpha\cos(\alpha),\ \alpha/\cos(\alpha)\\ \alpha+e^{\alpha},\ \alpha e^{\alpha},\ \alpha/e^{\alpha}$$ transcendental?