# Proving trigonometric equation $\cos(36^\circ) - \cos(72^\circ) = 1/2$ [duplicate]

$\cos \left( 36^\circ \right)-\cos \left( 72^\circ \right) = \frac{1}{2}$

Thank you.

## marked as duplicate by Martin Sleziak, Mauro ALLEGRANZA, Omran Kouba, Norbert, user91500Jun 9 '14 at 9:14

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:

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

• If you could give me the link for the trig functions at the given angles? Thank you. – xzatulx Jun 9 '14 at 5:24
• @xzatulx: you can type my username and scroll down the posts and find the one with "trig equations". – DeepSea Jun 9 '14 at 5:27
• Or, you can see this: cut-the-knot.org/pythagoras/cos36.shtml – JimmyK4542 Jun 9 '14 at 5:27