Consider a regular 10 - gon with its vertices on the unit circle. With one vertex fixed, draw straight lines to the other 9 vertices. Call them L 1 ,   L 2 , … , L 9 and denote their lengths by l 1 , l 2 , … , l 9 respectively. Then the product l 1 , l 2 , … , l 9 is

Step-by-step solution

Let α = e i 2 π 10 = e i π 5 Now, z 10 - 1 = ( z - 1 ) ( z - α ) … z - α 9 … . ( 1 ) so , ℓ 1 ℓ 2 … ℓ 9 = | 1 - α | 1 - α 2 … 1 - α 9 = ( 1 - α ) 1 - α 2 … 1 - α 9 = lim z → 1 z 10 - 1 z - 1 = 10