Question:
Write the value of cos 1° + cos 2° + cos 3° + ... + cos 180°.
Solution:
$\cos 1^{\circ}+\cos 2^{\circ}+\cos 3^{\circ}+\ldots+\cos 180^{\circ}$
$=\cos 1^{\circ}+\cos 2^{\circ}+\cos 3^{\circ}+\ldots+\cos 88^{\circ}+\cos 89^{\circ}+\cos 90^{\circ}+\cos (180-89)^{\circ}+\cos (180-88)^{\circ}+\ldots$$+\cos (180-1)^{\circ}+\cos 180^{\circ} \quad\left[\cos \left(180^{\circ}-\theta\right)=-\cos \theta\right]$
$=\cos 1^{\circ}+\cos 2^{\circ}+\cos 3^{\circ}+\ldots+\cos 88^{\circ}+\cos 89^{\circ}+\cos 90^{\circ}-\cos 89^{\circ}-\cos 88^{\circ}-\ldots-\cos 1^{\circ}+$$\cos 180^{\circ}$
$=\cos 90^{\circ}+\cos 180^{\circ}$
$=0-1$
$=-1$
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