How these 2 approximation equations are gained of euclild distances? 
$$r\gg 0$$
$$r_{1}\approx r-\frac{d}{2}\cos(\theta)\tag{1}$$
$$r_{2}\approx r+\frac{d}{2}\cos(\theta)\tag{2}$$
How the above approximation equations are made?

Use law of cosines and $d\ll r$
$r_1^2=r^2+(d/2)^2-dr\cos(\theta)$
$r_2^2=r^2+(d/2)^2+dr\cos(\theta)$
Divide by $r^2$ and get:
$(r_1/r)^2=1+(d/2r)^2-(d/r)\cos(\theta)$
Use $d\ll r$ and expand square root to first order
to get $r_1/r\approx 1-(d/2r)\cos(\theta)$ or $r_1\approx r-(d/2)\cos(\theta)$
Similarly  $r_2\approx r+(d/2)\cos(\theta)$
