i came up with the following: $$ |\vec{r}+\vec{s}|\cdot|\vec{r}-\vec{s}|=|\vec{r}^2-\vec{s}^2| $$ here is my calulation: $$ |\vec{r}+\vec{s}|\cdot|\vec{r}-\vec{s}|=\sqrt{(r_x+s_y)^2+(r_y+s_y)^2+(r_z+s_z)^2}\cdot\sqrt{(r_x-s_x)^2+(r_y-s_y)^2+(r_z-s_z)^2}=\sqrt{[(r_x+s_y)^2+(r_y+s_y)^2+(r_z+s_z)^2]\cdot[(r_x-s_x)^2+(r_y-s_y)^2+(r_z-s_z)^2]}=\sqrt{(\vec{r}+\vec{s})\cdot(\vec{r}-\vec{s})}=\sqrt{\vec{r}^2-\vec{s}^2} $$ but this cant be quite right as $|\vec{r}+\vec{s}|\cdot|\vec{r}-\vec{s}|$ cant be complex but $\sqrt{\vec{r}^2-\vec{s}^2}$ can be complex I dont know where the mistake is an i hope someone sees it
PS: in the last step I used $(\vec{r}+\vec{s})(\vec{r}-\vec{s})=\vec{r}^2-\vec{s}^2$