How do we prove $\cos(\pi/5) - \cos(2\pi/5) = 0.5$ ?. How do we prove $\cos(\pi/5) - \cos(2\pi/5) = 0.5$ without using a calculator.
Related question: how do we prove that $\cos(\pi/5)\cos(2\pi/5) = 0.25$, also without using a calculator

There is a neat trick:
\begin{align}
\cos 36^\circ - \cos72 ^\circ &=
(\cos^2 36^\circ - \cos^2 72 ^\circ) / (\cos 36^\circ + \cos72 ^\circ) \\
&= (\cos^2 36^\circ - \cos^2 72 ^\circ) / ((1 - 2\sin^2 18^\circ) + (2\cos^2 36^\circ - 1)) \\
&= (\cos^2 36^\circ - \cos^2 72 ^\circ) / 2(\cos^2 36^\circ - \sin^2 18^\circ) \\
&= (\cos^2 36^\circ - \cos^2 72 ^\circ) / 2(\cos^2 36^\circ - \cos^2 72^\circ) \\
&= 1/2
\end{align}
