Distance between Two Points in Plane in Polar Coordinates/Proof 1

Proof
Let $A$ and $B$ be embedded as suggested in a polar coordinate plane whose pole is at $O$.


 * Distance-polar-form.png

The distance $d$ is the side $AB$ of the triangle $AOB$.

We have that:
 * $OA = r_1$
 * $OB = r_2$

and:
 * $\theta_2 - \theta_1$ is the opposite angle to $AB$.

Hence we can use the Cosine Rule:
 * $AB^2 = r_1^2 + r_2^2 - 2 r_1 r_2 \map \cos {\theta_2 - \theta_1}$

From Cosine Function is Even we have that:
 * $\map \cos {\theta_2 - \theta_1} = \map \cos {\theta_1 - \theta_2}$

and the result follows.