How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$? I would like to find the apothem of a regular pentagon. It follows from $$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$ But how can this be proved (geometrically or trigonometrically)?

Since $x := \cos \frac{2 \pi}{5} = \frac{z + z^{-1}}{2}$ where $z:=e^{\frac{2 i \pi}{5}}$, and $1+z+z^2+z^3+z^4=0$ (for $z^5=1$ and $z \
eq 1$), $x^2+\frac{x}{2}-\frac{1}{4}=0$, and voilà.