Prove $\cos \frac{\pi}5-\cos \frac{2 \pi}5=\frac12$ but without finding $\cos \frac{ \pi}5$

Question

I can find the value of $\cos \left(\frac{ \pi}{5}\right)$, but is there a way to prove the equality without finding it?

I tried looking for both algebraic and geometric methods, but couldn't find anything

Answer 1

$$\cos36^{\circ}-\cos72^{\circ}=-\cos72^{\circ}-\cos144^{\circ}=-\frac{2\sin36^{\circ}\cos72^{\circ}+2\sin36^{\circ}\cos144^{\circ}}{2\sin36^{\circ}}=$$ $$=-\frac{\sin108^{\circ}-\sin36^{\circ}+\sin180^{\circ}-\sin108^{\circ}}{2\sin36^{\circ}}=\frac{1}{2}.$$