Question

Simplify: $\cos 1^{\circ }+\cos 2^{\circ }+\cos 3^{\circ }+....+\cos {180}^{\circ }$

Answer 1

$\cos 1^{\circ }+\cos 2^{\circ }+\cos 3^{\circ }+...\dots +\cos {177}^{\circ }+\cos {178}^{\circ }+\cos {179}^{\circ }+\cos {180}^{\circ }$

$=(\cos 1^{\circ }+\cos {179}^{\circ })+(\cos 2^{\circ }+\cos {178}^{\circ })+(\cos 3^{\circ }+\cos {177}^{\circ })+.....+\cos {180}^{\circ }$

We know that $\cos (180-x)=-\cos x$

$=(\cos 1^{\circ }-\cos 1^{\circ })+(\cos 2^{\circ }-\cos 2^{\circ })+(\cos 3^{\circ }-\cos 3^{\circ })+.....+\cos {180}^{\circ }$

$=0+0+0+...\dots .+(-1)$

$=-1$.

Hence, $\cos 1^{\circ }+\cos 2^{\circ }+\cos 3^{\circ }+....+\cos {180}^{\circ }=-1$