show that the area of a regular n-gon inscribed in a circle to the area of a regular n-gon circumscribing the same circle is cos^2(pi/n) : 1
area of triangular sections from each n-gon ...
red area = $\displaystyle \frac{1}{2}r^2 \sin\left(\frac{2\pi}{n}\right)$
red area + blue area = $\displaystyle r^2 \tan\left(\frac{\pi}{n}\right)$
ratio ...
$\displaystyle \frac{\sin\left(\frac{2\pi}{n}\right)}{\tan \left(\frac{\pi}{n}\right)}$
$\displaystyle \frac{2\sin\left(\frac{\pi}{n}\right) \cos\left(\frac{\pi}{n}\right)}{2\tan \left(\frac{\pi}{n}\right)}$
$\displaystyle \sin\left(\frac{\pi}{n}\right)\cos\left(\frac{\pi} {n}\right) \cdot \frac{\cos\left(\frac{\pi}{n}\right)}{\sin \left(\frac{\pi}{n}\right)}$
$\displaystyle \cos^2\left(\frac{\pi}{n}\right)$
look at the upper large right triangle (half of the entire large triangular section of the circumscribed n-gon)...
angle = $\displaystyle \frac{\pi}{n}$
adjacent side = $\displaystyle r$
opposite side = $\displaystyle r \tan\left(\frac{\pi}{n}\right)$
area of upper right triangle = $\displaystyle \frac{1}{2} \cdot r \cdot r \tan\left(\frac{\pi}{n}\right)$
double that ...