Let $p$ be an odd prime number. I want to compute the degree
$$
[\mathbb{Q}(\cos(2\pi/p),\sin(2\pi/p)):\mathbb{Q}(\cos(2\pi/p))].
$$
I already showed that
$$
[\mathbb{Q}(\cos(2\pi/p)):\mathbb{Q}] = \frac{p-1}{2},
$$
so it will be helpful to know
$$
[\mathbb{Q}(\cos(2\pi/p),\sin(2\pi/p)):\mathbb{Q}].
$$
We know that the polynomial
$$
X^2+\cos^2(2\pi/p)-1 \in \mathbb{Q}(\cos(2\pi/p))[X]
$$
vanish on $\sin(2\pi/p)$, but I don't how to show that this polynomial is irreducible over $\mathbb{Q}(\cos(2\pi/p))$. I believe that 
$$
[\mathbb{Q}(\cos(2\pi/p),\sin(2\pi/p)):\mathbb{Q}(\cos(2\pi/p))]=2,
$$
but I cannot see why it is possible or not to write
$$
\sin(2\pi/p) = \sum\_{k=0}^{n} a\_k\cos^k(2\pi/p)
$$
for some $a\_0,a\_1,\dots,a\_n\in\mathbb{Q}$. 

Any help will be appreciated!