Koundinya-Mathematics Teachers Hub: October 2020

$Simplify\,\,the\,\,expressions\,\,of\,\,the\,\,sums\\\\ (a)\,\,cot^2\frac{\pi}{2n+1}+cot^2\frac{2\pi}{2n+1}+cot^2\frac{3\pi}{2n+1}+...+cot^2\frac{n\pi}{2n+1}\\ (b)\,\,csc^2\frac{\pi}{2n+1}+csc^2\frac{2\pi}{2n+1}+csc^2\frac{3\pi}{2n+1}+...+csc^2\frac{n\pi}{2n+1}\\\\ from\,\,the\,\,above\,\,show\,\,that\,\,it\,\,follows\\\\ 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+......=\frac{\pi^2}{6}\\\\ where\,\,the\,\,sum\,\,of\,\,the\,\,infinite\,\,series\,\,1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+......\\is\,\,understood\,\,as\,\,the\,\,limit\,\,to\,\,which\,\,the\,\,finite\,\,sum\\\\ 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+......+\frac{1}{n^2}\,\,tends\,\,as\,\,n\to\infty.$

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