Question

Prove that 1/(4-sec^2⁡〖2π/7〗 )+1/(4-sec^2⁡〖4π/7〗 )+1/(4-sec^2⁡〖6π/7〗 )=1

Answer 1

Answer:

I tried doing this way:

Look at y=cosθ+isinθ where θ∈{2π7,4π7,⋯,2π}

Then we have

y7y7−1(y−1)⋅(y6+y5+⋯+1)=1=0=0

Now the root y=1 corresponds to θ=2π, and that

y6+y5+⋯+1=0

have roots cosθ+isinθ, where θ∈{2π7,4π7,⋯}. Looking at y+1y will give me the roots as cosθ and then i can put z=y2 to get cos2 as the roots and the invert to get sec2, but I have some problems.