Or, even more simple, in one of the right angle triangles you've seen, tan(theta/2)=L2/L1

I think also L1 definition is a little bit confusing, for me is just the bend radius (as length).

About your remark: yes, knowing more math the danger is to get complicated but in the end correctness of the result is what really matters.
Your result is correct and can be simplified as
1/sin(theta)-1/tan(theta)=(1-cos(theta))/sin(theta)=
=[2*sin^2(theta/2)]/[2*sin(theta/2)*cos(theta/2)]=
=sin(theta/2)/cos(theta/2)=
=tan(theta/2)