Relation between trace and norm in a quadratic field extension

Question

In the book of Y.Kitaoka about Arithmetic of quadratic forms page 12, I found these relations: for a finite field $F$ and by considering the quadratic extension $F':=F(\theta)$ $$tr(x\bar{y})=N(x+y)-N(x)-N(y)$$ $$\forall x\in \overline{F'},\quad tr\left(x\overline{F'}\right)=0\implies x=0$$ where $\bar{y}$ denotes the conjugate of $y$ over $F$, $N$ and $tr$ are the norm and trace mapping from $F'$ to $F$. So my question as follows; how to prove this equality and this implication ?. I tried to express: $$tr(x\bar{y})=\sum_{\sigma\in Gal(F'/F)}\sigma(x)\sigma(\bar{y})$$ and \begin{align*} N(x+y)&=\prod_{\sigma\in Gal(F'/F)}\big(\sigma(x)+\sigma(\bar{y})\big)\\ &=\prod_{\sigma\in Gal(F'/F)}\sigma(x)+\prod_{\sigma\in Gal(F'/F)}\sigma(\bar{y})+\prod_{\sigma,\tau\in Gal(F'/F)}\sigma(x)\tau(\bar{y})\\ &=N(x)+N(\bar{y})+\prod_{\sigma,\tau\in Gal(F'/F)}\sigma(x)\tau(\bar{y}) \end{align*}