Let $P(z)=\sum_{k=0}^n a_kz^k$ where $0＜a_1＜\cdots＜a_n$. Show that $P(z)$ has n zeros in unit disk $\Bbb D$. Moreover, show that $\sum_{k=0}^n a_kcoskt$ has 2n zeros in open interval $(0,2π)$.
I want to use Rouché theorem but I can't find a analytic function has n zero in $\Bbb D$.