Question

Prove that the sum of the radii of the circles, which are respectively inscribed and circumscribed about a regular polygon of n sides , is (a/2) cot ($\pi$/2n), where a is the side of the polygon

Answer 1

We have r + R $=\frac{a}{2}cot\frac{\pi }{n}+\frac{a}{2}cosec\frac{\pi }{n}$
$=\frac{a}{2}\left[\frac{cos\left(\pi /n\right)+1}{sin\left(\pi /n\right)}\right]=\frac{a}{2}\cdot \frac{2co{s}^2\left(\pi 2n\right)}{2sin\left(\pi /2n\right)cos\left(\pi /2\right)}$
$=\frac{a}{2}cot\frac{\pi }{2n}$