<s>[INST] 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
 [/INST] Here is a solution using only elementary trigonometric relationships.  We do not need complex variables or advance knowledge of the cosine values.
Let $x = \cos (\pi/5) - \cos (2\pi/5)$.  Compute $x^2$:
$$x^2 = \cos^2 (\pi/5) - 2 \cos (\pi/5)\cos (2\pi/5) + \cos^2 (2\pi/5)$$
Use the standard trigonometric identities:
\begin{split} \cos^2 (\pi/5) + \cos^2 (2\pi/5) &= \frac{1 + \cos (2\pi/5)}{2}
    + \frac{1 + \cos (4\pi/5)}{2} \\
&= 1 - \frac 12(\cos (\pi/5) - \cos (2\pi/5))\\
&= 1 - \frac x2\end{split}
and 
\begin{split} \cos (\pi/5) \cos (2\pi/5) &= \frac 12(\cos (3\pi/5) + \cos(-\pi/5))\\
&= \frac 12 (\cos (\pi/5) - \cos (2\pi/5))\\
&= \frac x2 
\end{split}
Then $x^2 = (1 - \frac x2 ) - 2(\frac x2)$, hence
$$2x^2 - 3x -2 = 0.$$
Thus $x = \cos (\pi/5) - \cos (2\pi/5) =1/2$, the only positive root of the above quadratic equation (clearly $x$ is positive).  Also the product of the cosines is $x/2 = 1/4$.
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