Hey guys, I am new to this forum and I hope to maybe become a regular user.

The following problem is actually from my high school investigation coursework, but it seems too hard to belong to the pre-college forum. By investigating the product of line segments between points of roots of unity, I managed to form the flowing conjecture:

n = integer greater than or equal to 2

I tried to prove it by induction, but I don't know how with this type of trigonometric function. By any method, can someone help me prove it?

EDIT: sorry, I mislabeled the thread. This is not actually a geometric progression even though is kind of like one

ALSO, even though this is part of schoolwork, my teacher stated that outside research and discussion is allowed, even among classmates doing the same investigation. As long as we understand why and how the proof works, we are not considered cheating if we learn do the question ourselves, so feel free to provide as much help as you want without feeling like your helping me cheat.