Could someone explain to me why $\tan\theta\approx y/s$? I can follow the other approximations, but here I don't see it. I'm guessing we have similar triangles here, but I wouldn't really know how to show it. It seems to me that I have to make one approximation first before showing the triangles are similar. Also, note: it's only given that we have $\theta_m$ in the triangle $(S_1,S_2,B)$. The other two $\theta_m$ are exactly the angles I try to show.
My suggestion is not to follow the approximations in the textbooks but rather do it in a rigorous way yourself. So we have
$$r_1-r_2=\sqrt{s^2+(y+a/2)^2}-\sqrt{s^2+(y-a/2)^2}$$
For $s>>y,a$, we have
$$r_1-r_2\approx s\left[1+\frac{1}{2}\frac{(y+a/2)^2}{s^2}-1-\frac{1}{2}\frac{(y-a/2)^2}{s^2}\right]$$ $$=\frac{(y+a/2)^2-(y-a/2)^2}{2s}$$ $$=\frac{ay}{s}$$
which is the final result you want.