\begin{align} \mathbf{C}_k\cdot \mathbf{C}_l & = \sum_{n=1}^N \cos\left(\frac{2\pi k n}{N} \right) \cos\left(\frac{2\pi l n}{N} \right)\\ &= \sum_{n=1}^N \frac{1}{2} \left( \cos\left(\frac{2\pi n}{N} (k+l)\right) + \cos\left(\frac{2\pi n}{N} (k-l)\right) \right)\\ &= \frac{N}{2}\, \delta(k-l) \\ \mathbf{S}_k\cdot \mathbf{S}_l &= \frac{N}{2}\,\delta(k-l)\\ \mathbf{C}_k\cdot \mathbf{S}_l &= 0\\ \end{align}
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