Let $G$ be a group of order $2n$ and $a,t\in G$ s.t $|a|=n, |t|=2$ and $tat=a^{-1}$. define $N=\langle a\rangle$.

I need to show that $|G:N|=2$ and $G=N\cup tN$.

to show that $|G:N|=2$ is trivial using Lagrange's Theorem but for showing that $G=N\cup tN$,$~~~$ I need to show that $t\not \in N$. how can I prove this?