$\cos^2\left(\frac{2k\pi}5\right)+\cos^2\left(\frac{4k\pi}5\right)= c$?

Question

I just came across this:

$$\cos^2\left(\frac{2k\pi}5\right)+\cos^2\left(\frac{4k\pi}5\right)= c$$ where $c$ is a positive, integer constant.

How do I obtain this? And what is the constant?

Thanks!

Answer 1

This can't be right: for k = 0 and k = 1 you get different values.

Answer 2

Using Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$

with $5A=2k\pi,5\nmid k\implies\sin5A\sin A\ne0,$

$6A=2k\pi+A,4A=2k\pi-A$

$$\cos^22A+\cos^24A=\cos^22A-\sin^24A+1$$ $$=1+\cos6A\cos2A=1+\cos A\cos2A=1+\dfrac{\sin2A\cos2A}{2\sin A}=1+\dfrac{\sin4A}{4\sin A}=1-\dfrac14$$