Let $p$ be an odd prime. For which values of $p$ is the following true: $$\mathbb{Q}\left(\sin\left(2\pi/p\right),\cos\left(2\pi/p\right)\right)=\mathbb{Q}\left(\sin\left(2\pi/p\right)\right),$$ where those are field extensions over $\mathbb{Q}$.
From $$p-1=[\mathbb{Q}(e^{2\pi/p}):\mathbb{Q}]=[\mathbb{Q}(e^{2\pi/p}):\mathbb{Q}\left(\sin\left(2\pi/p\right),\cos\left(2\pi/p\right)\right)]\cdot[\mathbb{Q}\left(\sin\left(2\pi/p\right),\cos\left(2\pi/p\right):\mathbb{Q}\right)]$$
I get $$[\mathbb{Q}\left(\sin\left(2\pi/p\right),\cos\left(2\pi/p\right)\right):\mathbb{Q}]=\frac{p-1}{2}.$$
But then $$[\mathbb{Q}\left(\sin\left(2\pi/p\right),\cos\left(2\pi/p\right)\right):\mathbb{Q}]=[\mathbb{Q}\left(\sin\left(2\pi/p\right),\cos\left(2\pi/p\right):\mathbb{Q}\sin(2\pi/p)\right)]\cdot[\mathbb{Q}\left(\sin\left(2\pi/p\right)\right):\mathbb{Q}]$$ $$\Rightarrow \frac{p-1}{2}=[\mathbb{Q}\left(\sin\left(2\pi/p\right),\cos\left(2\pi/p\right):\mathbb{Q}\sin(2\pi/p)\right)].[\mathbb{Q}\left(\sin\left(2\pi/p\right)\right):\mathbb{Q}]$$
I came across the following article What is $\operatorname{Gal}(\mathbb Q(\zeta_n)/\mathbb Q(\sin(2\pi k/n))$? which states that (since clearly $p$ prime $\Rightarrow p$ is not $0\mod4$) $$[\mathbb{Q}(\sin(2π/p)):\mathbb{Q}]=ϕ(p)=p-1$$
which forces $$[\mathbb{Q}\left(\sin\left(2\pi/p\right),\cos\left(2\pi/p\right):\mathbb Q(\sin(2\pi/p)\right)]=\frac{1}{2}.$$
Clearly something went wrong but I'm not sure what - I have a feeling I'm making some elementary error.