If a regular polygon of z sides has the circum radius -E and inradius r and has its side equal to $2a$. Then $a\cot \frac{\pi }{2n}$ is

Here side $A_1{A}_2=2a\Rightarrow A_{2}D=a=A_1,D$
 

$\begin{array}{l}\mathrm{\angle }{A}_{1}OD=\mathrm{\angle }{A}_{2}OD=\alpha =\frac{\pi }{n}\\ \therefore r=a\cot \frac{\pi }{n},R=a\mathrm{cosec}\frac{\pi }{n}\\ \therefore r+R=a\left[\cot \frac{\pi }{n}+\mathrm{cosec}\frac{\pi }{n}\right]\\ =a\left[\frac{1+\cos \frac{\pi }{n}}{\sin \frac{\pi }{n}}\right)=a\cdot \frac{2{\cos }^2\frac{\pi }{2n}}{2\sin \frac{\pi }{2n}\cos \frac{\pi }{2n}}\\ \Rightarrow r+R=a\cot \frac{\pi }{2n}\end{array}$