Prove that cos (2π/13)+ cos (6π/13) + cos (8π/13)= (sqrt(13)-1)/4
Outline proof: For , let denote the angle . The points are equally spaced around the unit circle, so their centre of mass is at the origin, and it follows that
Since and , it follows that
Next,
.
But (with similar results for and ). Also, (with similar results for the other two products and ).
Putting all that together, you find that
, and
Now let and It follows from the above calculations that and Eliminate T between those equations to get , a quadratic with solutions Now all that remains for you to do is to figure out why the square root must have the plus sign.
The quantities S and T are the two roots of the quadratic equation. It should be clear from a diagram of the 13 points on the unit circle that the three cosines making up the sum T are smaller than their counterparts in S. Specifically, , and . Therefore T<S and so S must correspond to the larger root of the quadratic.