cos⁡1∘+cos⁡2∘+cos⁡3∘+….+cos⁡179∘=

cos⁡1∘+cos⁡2∘+cos⁡3∘+…+cos⁡179∘=cos⁡1∘+cos⁡179∘+cos⁡2∘+cos⁡178∘+…+cos⁡89∘+cos⁡91∘+cos⁡90∘=cos⁡1∘−cos⁡1∘+cos⁡2∘−cos⁡2∘+…+cos⁡89∘−cos⁡89∘+cos⁡90∘=0

cos⁡1∘+cos⁡2∘+cos⁡3∘+….+cos⁡179∘=

cos⁡1∘+cos⁡2∘+cos⁡3∘+…+cos⁡179∘=cos⁡1∘+cos⁡179∘+cos⁡2∘+cos⁡178∘+…+cos⁡89∘+cos⁡91∘+cos⁡90∘=cos⁡1∘−cos⁡1∘+cos⁡2∘−cos⁡2∘+…+cos⁡89∘−cos⁡89∘+cos⁡90∘=0

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

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Detailed Solution

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