Verified answer

distance^2:=s^2=(x1-x2)^2+(y1-y2)^2

but:

x1=r1*cos(θ1), y1=r1*sin(θ1)

x2=r2*cos(θ2), y2=r2*sin(θ2)

so

1) // substitute:

s^2=[r1*cos(θ1)-r2*cos(θ2)]^2 +[r1*sin(θ1)-r2*sin(θ2) ]^2=

2) //expand the terms in parentheses, and add together

= r1^2*(cos(θ1)^2+sin(θ1)^2) +r2^2*(cos(θ2)^2+sin(θ2)^2) -2*r1*r2*cos(θ1)cos(θ2)-

2* r1*r2* sin(θ1)*sin(θ2)=

3) //use the given identity and add the last two terms together

= r1^2+r2^2 -2*r1*r2*(cos(θ1)cos(θ2)+sin(θ1)sin(θ2)=

4) // use the identity cos(θ1-θ2)=cos(θ1)cos(θ2)+sin(θ1)sin(θ2)

= r1^2+r2^2-2*r1*r2*cos(θ1-θ2)

and so

s=sqrt(r1^2+r2^2-2*r1*r2*cos(θ1-θ2) ) ,

which is just the law of cosines ;)