Proving trigonometric equation $\cos(36^\circ) – \cos(72^\circ) = 1/2$

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As $\displaystyle\cos(180^\circ-y)=-\cos y$

$$S=\cos36^\circ-\cos72^\circ=\cos36^\circ+\cos(180^\circ-108^\circ)=\cos36^\circ+\cos108^\circ$$

Now multiplying the numerator & the denominator by $\displaystyle2\sin\frac{(108-36)}2^\circ,$

$$S=\frac{2\sin36^\circ\cos36^\circ+2\sin36^\circ\cos108^\circ}{2\sin36^\circ}$$

Using Werner Formula & $\sin2x$ formula,
$$S=\frac{\sin72^\circ+\sin144^\circ-\sin72^\circ}{2\sin36^\circ}=\frac{\sin(180^\circ-36^\circ)}{2\sin36^\circ}$$

Hope you can take it home from here

From a recent post which relates to the above trig functions at the given angles, we have:

$\cos36^{\circ} = \sin54^{\circ} = \dfrac{\sqrt{5}+1}{4}$, and:

$\cos72^{\circ} = \sin18^{\circ} = \dfrac{\sqrt{5}-1}{4}$. Thus:

$\cos36^{\circ} – \cos72^{\circ} = \dfrac{\sqrt{5}+1}{4} – \dfrac{\sqrt{5}-1}{4} = \dfrac{1}{2}$