During my learning process, I encountered one of the following expressions. $$ 2 \cos \left(\frac{\pi }{k+2}\right) \prod _{j=1}^{2 k+3} \left(2 \cos \left(\frac{\pi }{k+2}\right)-2 \cos \left(\frac{2 j \pi }{2 k+3}\right)\right)-\prod _{j=1}^{2 k+2} \left(2 \cos \left(\frac{\pi }{k+2}\right)-2 \cos \left(\frac{j \pi }{2 k+3}\right)\right) $$ I guess this value is greater than 0 when k takes any positive integer. I used math software to verify some values. Fortunately, there was no counterexample.. But the specific mathematical proof, I did not think of a good way.
G[k_] := 2 Cos[Pi/(k + 2)]*
Product[(2 Cos[Pi/(k + 2)] - 2 Cos[2 j*Pi/(2 k + 3)]), {j, 1,
2 k + 3}] -
Product[(2 Cos[Pi/(k + 2)] - 2 Cos[j*Pi/(2 k + 3)]), {j, 1,
2 k + 2}] // N
ListPlot[Table[G[k], {k, 1, 100}]]
