Question: Given the equations (1):$$\frac{\partial u}{\partial t}+g\frac{\partial \eta}{\partial x}=0$$ 
and (2):$$\frac{\partial\eta}{\partial t}+H\frac{\partial u}{\partial x}=0$$
can we combine the two together 
to form $$\frac{\partial^{2}\eta}{\partial t^{2}}-gH\frac{\partial^{2}\eta}{\partial x^{2}}=
0$$ by substituting $\frac{\partial}{\partial t}.(2)$ into (1). 
I should note this was given as an example but the working was not 
shown, I have tried a few manipulations but cannot get the result, 
am I missing something?

Answer: 
(1) to show that $\cos(\frac {m}{n} \pi)$ is algebraic it is enough to show that 
$\cos(\frac{1}{n} 2\pi)$ is algebraic. So it comes down to showing that the real part of an $n$-th root of unity is algebraic.

(3) there is a transcendental $a$ not equal to any rational multiple of $\pi$. Just take square root of $(\pi -1)$, lest $\pi$ would be algebraic.
(4) $\cos(a\pi)=\frac{e^{ia\pi}+e^{-ia\pi}}{2}$ so $\cos(a\pi)$ is algebraic iff $e^{ia\pi}$ is algebraic.

Since $e^{ia\pi} = (e^{i\pi})^a$ and $e^{i\pi}=-1$ is algebraic, the [Gelfond-Schneider theorem](https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem) implies that $\cos(a\pi)$ is transcendental if $a$ is algebraic but not rational.