Let $n$ be a nonnegative integer and $a_{0}, a_{1}, ..., a_{n}$ real numbers. For any real number $t$ let $f(t)= \sum_{k=0}^{n}a_{k}\cos(kt)$. Could you help me with the following two questions ?
a) Prove that there exists a polynomial $P \in \mathbb{C}[X]$ such that for all real $t$ : $f(t)=e^{-int}P(e^{it})$.
b)Suppose that for all real $t$ $f(t)=0$. Prove that all $a_{k}$s are equal to zero.