My teacher gave me this problem as a personal homework. I need to determine if $\cos(2\pi/13)$ and $\cos(2\pi/55)\in K = \mathbb{Q}[\cos(2\pi/37),\cos(2\pi/15),\cos(2\pi/11)]$.
For $\cos(2\pi/13$) I said that $\mathbb{Q}[\cos(2\pi/13)] \subset \mathbb{Q}[\zeta_{13}]$ and $K \subset \mathbb{Q}[\zeta_{37*15*11}]$ and we know that $\mathbb{Q}[\zeta_{m}] \cap \mathbb{Q}[\zeta_{n}] = \mathbb{Q}$ when gcd(m,n) = 1. So $cos(2\pi/13) \notin K$.
But that is not working for $\cos(2\pi/55)$. So I came up with another idea. Let $G(\mathbb{Q}[\zeta_{37*15*11}]/\mathbb{Q}) = \{\sigma_i \mid \sigma_i : \omega \to \omega^{i}\text{ for }\gcd(i,37*15*11) = 1\}\simeq \mathbb{Z}_{37*15*11}^*$. I want to look at $G(\mathbb{Q}[\zeta_{37*15*11}]/K)$ and $G(\mathbb{Q}[\zeta_{37*15*11}]/\mathbb{Q}[\cos(2\pi/55)])$ and compare them. That's where I am stuck. I know that I need to find such elements in $G(\mathbb{Q}[\zeta_{37*15*11}]/\mathbb{Q})$ that stabilized $K$ and $\mathbb{Q}[\cos(2\pi/55)])$. But I don't understand how to do that.
Basically, the question is how $G(\mathbb{Q}[\zeta_{37*15*11}]/K)$ and $G(\mathbb{Q}[\zeta_{37*15*11}]/\mathbb{Q}[\cos(2\pi/55)])$ looks like and will it give me the solution of my problem.
Your argument for $\cos(2\pi/13)$ is fine (as $\cos(2\pi/13)\not \in \Bbb{Q}=\Bbb{Q}(\zeta_{11\cdot 15\cdot 37})\cap \Bbb{Q}(\zeta_{13})$).
Then look at the automorphism of $\Bbb{Q}(\zeta_{11\cdot 15\cdot 37})$ sending $\zeta_{11\cdot 15 \cdot 37}$ to $ \zeta_{11\cdot 15 \cdot 37}^n$, where $ n\equiv -1\bmod 15,n\equiv 1\bmod 11,n\equiv 1\bmod 37$. Does it fix $\cos(2\pi/37),\cos(2\pi/15),\cos(2\pi/11),\cos(2\pi/55)$ ?