Evaluate: \(\sum _{r=1}^{13}\frac{1}{\sin \left[\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right]\sin \left[\frac{\pi }{4}+\frac…

\(\sum _{r=1}^{13}\frac{1}{\sin \left[\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right]\sin \left[\frac{\pi }{4}+\frac{r\pi }{6}\right]}\\ \sum _{r=1}^{13}\frac{\sin \left[\frac{\pi }{4}+r\frac{\pi }{6}\right]-\left[\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right]}{\left(\sin \frac{\pi }{6}\right)\times \sin \left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{r\pi }{6}\right)}\\ \sum _{r=1}^{13}2\left(\cot \left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)-\cot \left(\frac{\pi }{4}+\frac{r\pi }{6}\right)\right)\\ 2\left[\cot \frac{\pi }{4}-\cot \left(\frac{\pi }{4}+\frac{13\pi }{6}\right)\right]\\ 2\left(1-\cot \frac{5\pi }{12}\right)\\ 2(1-2+\sqrt{3})\Rightarrow (2\sqrt{3}-2)\)