Approximating difference between sides of triangle

Using the cosine rule for the triangle on the right hand side

$R^2_2=r^2+\frac{d^2}{4}-rd\cdot cos\theta$

And using the cosine rule on the triangle on the left side

$R^2_1=r^2+\frac{d^2}{4}-rd\cdot cos(180-\theta )$

$R^2_1=r^2+\frac{d^2}{4}-rd\cdot (cos(180)cos(\theta)+sin(180sin(\theta))$

cos180= -1, sin180=0

$R^2_1=r^2+\frac{d^2}{4}+rd\cdot cos\theta$

$R^2_2-R^2_1= r^2+\frac{d^2}{4}+rd\cdot cos\theta -(r^2+\frac{d^2}{4}-rd\cdot cos\theta)$

$R^2_2-R^2_1= 2rd\cdot cos\theta$

$(R_2-R_1)(R_2+R_1)= 2rd\cdot cos\theta$

As r becomes much larger than d R1 and R2 become nearly equal to r

$R_1+R_2 \approx 2r$

$(R_2-R_1)\frac{(R_2+R_1)}{2r}= d\cdot cos\theta$

$R_2-R_1= d\cdot cos\theta$