The Sum of Cosines

Relevant wiki: Sine and Cosine Graphs

$cos (180-n)° = -cos (n)°$

From this point, we can regroup each pair (i.e. ${1;179},{2;178}$ ) that will add up to 0. Knowing that $cos (90)° = 0$ and will be left alone, our sum equals $89\times0 + 0 = 0$ .

Let the sum of cosines be $S$ . Regroup them such that $S=(\cos{1^{\circ}}+\cos{179^{\circ}})+(\cos{2^{\circ}}+\cos{178^{\circ}})+ \cdots + (\cos{89^{\circ}}+\cos{91^{\circ}}) + \cos{90^{\circ}}$ Note that $\cos A + \cos (180^{\circ}-A) = \cos A + \cos 180^{\circ} \cos A + \sin 180^{\circ} \sin A = \cos A - \cos A = 0$ . Therefore $S=(0)+(0)+ \cdots + (0) + 0 = \large \color{#20A900}{\boxed{0}}.$

Geometrically it's like a summa of The positive and negative abscissas of The Ray of a circle (The goniometriche circle) for The projection of every angle from 0 to π which simmetrically gives 0.