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