Prove that the sum of the radii of the circles, which are, respectively, inscribed and circumscribed about a polygon of `n` sides, whose side length is `a ,` is `a/2 `cot`pi/(2n)`

Explanation

Radius of the circumscribed circle `= R = (a)/(2) cosec.(pi)/(n)`
And, radius of the inscribed circle `= r = (1)/(2) a cot ((pi)/(n))`
`rArr R + r = (a)/(2 sin (pi//n)) + (a cos (pi//n))/(2 sin (pi//n))`
`= (a[1 + cos (pi//n)])/(2 xx 2 sin (pi//2n) cos (pi//2n))`
`= (2a cos^(2).(pi)/(2n))/(4 sin.(pi)/(2n) cos.(pi)/(2n))`
`= (1)/(2) a cot ((pi)/(2n))`