The value of $\sum _{k=1}^{13}\frac{1}{\sin \left(\frac{\pi }{4}+\frac{\left(k-1\right)\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{k\pi }{6}\right)}$ is equal to

Let \(\frac{\pi}{4}+(K-1) \frac{\pi}{6}=\theta\)
\(\Rightarrow \sum_{k=1}^{13} \frac{1}{\sin \theta \sin \left(\theta+\frac{\pi}{6}\right)}=\frac{1}{\sin \frac{\pi}{6}} \sum_{k=1}^{13}\) \(\frac{\sin ^\pi / 6}{\sin \theta \sin \left(\theta+{ }^\pi / 6\right)}\)
\(=2 \sum_{k=1}^{13} \frac{\sin \left(\theta+\frac{\pi}{6}-\theta\right)}{\sin \theta \sin \left(\theta+\frac{\pi}{6}\right)}=2 \sum_{k=1}^{13}\) \(\frac{\sin \left(\theta+{ }^\pi / 6\right) \cos \theta-\cos \left(\theta+{ }^\pi / 6\right) \sin \theta}{\sin \theta \sin \left(\theta+{ }^\pi / 6\right)}\)
\(=2 \sum_{k=1}^{13}\left(\cot \theta-\cot \left(\theta+\frac{\pi}{6}\right)\right)=2 \sum_{k=1}^{13}\)\(\left(\cot \left(\frac{\pi}{4}+(k-1) \frac{\pi}{6}\right)-\cot \left(\frac{\pi}{4}+\frac{k \pi}{6}\right)\right)\)
\(=2(\cot \frac{\pi}{4}-\cot \left(\frac{\pi}{4}+\frac{\pi}{6}\right)+\cot \left(\frac{\pi}{4}+\frac{\pi}{6}\right)\) \(-\cot \left(\frac{\pi}{4}+\frac{2 \pi}{6}\right)+-----\cot \left(\frac{\pi}{4}+\frac{12 \pi}{6}\right)-\) \(\cot \left(\frac{\pi}{4}+\frac{13 \pi}{6}\right)) \)
\( =2\left[\cot \frac{\pi}{4}-\cot \left(\frac{\pi}{4}+\frac{13 \pi}{6}\right)\right]=2\left[1-\cot \frac{29 \pi}{12}\right]\) \(=2\left[1-\cot 75^{\circ}\right]=2[1-(2-\sqrt{3})] \)
\( =2(\sqrt{3-1})\)