A great Sum

The given sum can be expressed as follows:

$\begin{aligned} S & = \sum_{k=0}^{17} \cos (20k+15)^\circ \\ & = \underbrace{\sum_{k=0}^8 \cos (20k+15)^\circ}_{\cos 15^\circ \cdots \cos 175^\circ} + \underbrace{\sum_{k=9}^{17} \cos (20k+15)^\circ}_{\cos 195^\circ \cdots \cos 355^\circ} \\ & = \sum_{k=0}^8 \cos (20k+15)^\circ \color{#3D99F6} + \sum_{k=0}^8 \cos (20k+195)^\circ & \small \color{#3D99F6} \text{Note that }\cos (180^\circ - \theta) = - \cos \theta \\ & = \sum_{k=0}^8 \cos (20k+15)^\circ \color{#3D99F6} - \sum_{k=0}^8 \cos (-20k-15)^\circ & \small \color{#3D99F6} \text{also }\cos (- \theta) = \cos \theta \\ & = \sum_{k=0}^8 \cos (20k+15)^\circ \color{#3D99F6}- \sum_{k=0}^8 \cos (20k+15)^\circ \\ & = \boxed 0 \end{aligned}$