If$\sum _{r=1}^{13}\left{\frac{1}{\sin \left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{r\pi }{6}\right)}\right}=a\sqrt{3}+b,a,b\in Z$, then $a^2+b^2$ is equal to : [JEE Main 2025, 28 Jan (Shift 2)]

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YESPYQ · Accepted Answer

Correct answer: d) 8. Let $S=\sum _{r=1}^{13}\frac{1}{\sin \left(\frac{\pi }{4}+\frac{r\pi }{6}\right)\sin \left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)}$ $=2\sum _{r=1}^{13}\frac{\sin \left(\frac{\pi }{4}+\frac{r\pi }{6}\right)-\left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)}{\sin \left(\frac{\pi }{4}+\frac{\pi }{6}\right)\sin \left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)}$ $=2\sum _{r=1}^{13}\left(\cot \left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)-\cot \left(\frac{\pi }{4}+\frac{\mathrm{r}\pi }{6}\right)\right)$ $=2\left[\cot \left(\frac{\pi }{4}\right)-\cot \left(\frac{\pi }{4}+\frac{13\pi }{6}\right)\right]$ $=2\left[1-\cot \left(\frac{\pi }{4}+2\pi +\frac{\pi }{6}\right)\right]$ $=2\left[1-\cot \left(\frac{5\pi }{12}\right)\right]=2\left(1-2+\sqrt{3}\right)$ $=2\sqrt{3}-2=a\sqrt{3}+b$ ${a}^{2}+{b}^{2}=8$

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

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