$\cos 1^{\circ }+\cos 2^{\circ }+\cos 3^{\circ }+\dots .+\cos {179}^{\circ }=$

$\begin{array}{l}\cos 1^{\circ }+\cos 2^{\circ }+\cos 3^{\circ }+\dots +\cos {179}^{\circ }\\ =\left(\cos 1^{\circ }+\cos {179}^{\circ }\right)+\left(\cos 2^{\circ }+\cos {178}^{\circ }\right)+\dots +\left(\cos {89}^{\circ }+\cos {91}^{\circ }\right)+\cos {90}^{\circ }\\ =\left(\cos 1^{\circ }-\cos 1^{\circ }\right)+\left(\cos 2^{\circ }-\cos 2^{\circ }\right)+\dots +\left(\cos {89}^{\circ }-\cos {89}^{\circ }\right)+\cos {90}^{\circ }=0\end{array}$