[tex]\displaystyle \sf \boxed{\begin{matrix}\text{F\'ormula de Werner} : \\\\\ \displaystyle \sf cos(a)+cos(b) = 2\cdot cos\left(\frac{a+b}{2}\right)\cdot cos\left(\frac{a-b}{2}\right) \end{matrix}}[/tex]

temos :

[tex]\displaystyle \sf S = cos(1^\circ)+cos(2^\circ)+cos(3^\circ)+...+cos(180^\circ) \\\\ \text{Observe que}: \\\\\ cos(179^\circ)+cos(1^\circ) = 2\cdot \underbrace{\sf cos(90^\circ)}_{0}\cdot cos(89^\circ) = 0 \\\\ cos(178^\circ)+cos(2^\circ) = 2\cdot \underbrace{\sf cos(90^\circ)}_{0}\cdot cos(88^\circ) = 0 \\\\ cos(177^\circ)+cos(3^\circ)=2\cdot \underbrace{\sf cos(90^\circ)}_{0}\cdot cos(87^\circ)=0 \\ ... \\ cos(89^\circ)+cos(91^\circ) =2\cdot \underbrace{\sf cos(90^\circ)}_{0}\cdot cos(1^\circ)=0 \\\\[/tex]
sobrando somente cos(90º) e cos(180º). Daí :

[tex]\sf S = 0+0+0+...+cos(90^\circ)+0+0+0+...+cos(180^\circ) \\\\ S = cos(90^\circ)+cos(180^\circ) \\\\ S = 0-1 \\\\ \Large\boxed{\sf \ S =- 1 \ }\checkmark[/tex]

outra solução
Existe uma fórmula para se calcular soma de cossenos cujo os ângulos estão em P.A, que é dada por :

[tex]\displaystyle \sf S= cos(\alpha)+cos(\alpha+r)+cos(\alpha+2r)+...+cos(\alpha+(n-1)r) \\\\\\\ \boxed{\sf \ S =cossec\left(\frac{r}{2}\right)\cdot cos\left(\alpha+\frac{(n-1)\cdot r}{2}\right)\cdot sen\left(\frac{n\cdot r}{2}\right)\ }[/tex]

temos :

[tex]\displaystyle \sf S = cos(1^\circ)+cos(2^\circ)+cos(3^\circ)+...+cos(180^\circ) \\\\ r = 1 \ ;\ n = 180\\\\\ \text{Substituindo na f\'ormula} : \\\\ S = cossec\left(\frac{1}{2}\right)\cdot cos\left(1+\frac{(180-1)\cdot 1}{2}\right)\cdot sen\left(\frac{180\cdot 1}{2}\right) \\\\\\ S = cossec\left(\frac{1}{2}\right)\cdot cos\left(\frac{2+179\cdot 1}{2}\right)\cdot \underbrace{\sf sen\left(90^\circ \right)}_{1} \\\\\\ S = cossec\left(\frac{1}{2}\right)\cdot cos\left(\frac{1+180}{2}\right)[/tex]

[tex]\displaystyle \sf S = cossec\left(\frac{1}{2}\right)\cdot cos\left(90^\circ +\frac{1}{2}\right) \\\\\\ \boxed{\begin{matrix}\sf obs: \\\\\ \displaystyle \sf cos\left(90+\frac{1}{2}\right ) = -sen\left(\frac{1}{2}\right) \end{matrix}} \\\\\\ Da{\'i}} : \\\\ S = cossec\left(\frac{1}{2}\right)\cdot \left[-sen\left(\frac{1}{2}\right)\right][/tex]

[tex]\displaystyle \sf S = \frac{\displaystyle -sen\left(\frac{1}{2}\right)}{\displaystyle sen\left(\frac{1}{2}\right)} = - 1 \\\\\ Portanto : \\\\ \Large\boxed{\sf \ S = - 1 \ }\checkmark[/tex]