Let $n>4$, and $(h,n) = 1$. How to show that $[\mathbb{Q}(\tan 2 \pi h/n):\mathbb{Q}]= \phi(n)$ or $\phi(n)/2$ or $\phi(n)/4$ respectively if $\gcd(n,8)<4$ or $\gcd(n,8)=4$ or $\gcd(n,8)>4$.
(In fact, this is a question from J. McCarthy's Algebraic Extensions of Fields, Ch. 2. We know from a previous question that $[\mathbb{Q}(\cos 2\pi h/n):\mathbb{Q}]=\phi(n)/2$ if $n>2$ and $\gcd(n,h)=1$; and also that if $n>4,$ $[\mathbb{Q}(\sin 2\pi h/n):\mathbb{Q}]=\phi(n), \phi(n)/4$ or $\phi(n)/2$, respectively if $\gcd(n,8)<4$, $\gcd(n,8)=4$ or $\gcd(n,8)>4$.)