How can I prove that for every $n \in \mathbb{N}$ there exist polynomials $p_n(x,y)$ and $q_n(x,y)$ in two variables $x,y$ with real coefficients such that:
$$\sin(nt) = p_n(\sin(t),\cos(t))\quad\text{and}\quad \cos(nt) = q_n(\sin (t), \cos(t))$$
for all $t \in \mathbb{R}$, using the Addition Theorem for $\sin$ and $\cos$?
Edit of my thoughts:
I know that this is obvious for n=1 but I'm not sure if I can use induction or rather how I can combine that with the Theorem. One of the answers already statet that I can indeed use induction so I will continue to try it that way.
Best Answer
We know that $\sin(a+t)=\sin(a)\cos(t)+\sin(t)\cos(a)$, and a similar formula for $\cos(a+b)$. Using this identity, we can prove the claim by induction on $n$, as follows.
When $n=1$, this is clear. If you assume this claim holds for both $\sin$ and $\cos$ for $n=m-1$, then you use the formula above with $a=(m-1)t$, to see that $\sin(mt)=\sin((m-1)t)\cos(t)+\sin(t)\cos((m-1)t)$. Since by induction $\sin((m-1)t)$ is a polynomial function in $\sin(t)$ and $\cos(t)$, then $\sin(mt)$ is a polynomial function. The same is true for $\cos(mt)$. Therefore, the claim holds for $n=m$.