Gauss' trigonometric identities for heptadecagon

Today, we write down Gauss' magical trigonometric identities for regular heptadecagon.

Construction algorithm

In our previous post, we have learnt about Similar Triangles and the Right Triangle Altitude Theorem. Today, continuing our journey in the garden of geometry, we want to find answer to the following question

Given a line segment of length $r$, by compass and straightedge, what kind of shapes can we construct? 

Construction of regular polygons

There are a few classic problems of ancient mathematics that are easy to state but incredibly difficult to solve. Take for example, the problem of constructing regular polygons and the problem of trisecting an angle by compass and straightedge. It was not until the 18th-19th centuries that mathematicians could finally solve them by employing advanced tools of number theory and algebra. Today, we will look at the regular polygon construction problem.

Regular polygon construction problem. Using compass and straightedge, construct a regular polygon with $n$ sides.