$$\frac {\displaystyle\sum\limits_{k=0}^{6} \operatorname{cosec}^{2}(\theta+\frac{k\pi}{7})}{7\operatorname{cosec}^{2}(7\theta)} = ?$$ Here, $\theta = \frac{\pi}{8}$. This question is from a book recommended for JEE Advance exam. I tried converting the series like, $$\frac{1}{\sin^{2}(\theta+\frac{k\pi}{7})}=\frac{2}{1-\cos(2\theta+\frac{2k\pi}{7})}$$ But this didn't lead to anything. Then I also tried taking the $k=0$ term out and pair up first and last terms which still didn't lead to anything. Hints or help would be highly appreciated.
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Hint: Make it telescoping. – An_Elephant May 04 '23 at 15:53
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I tried to do that by turning it into cot but that also didn't help me. So could you please elaborate? – Gurasees May 04 '23 at 15:56
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It's a book provided by my coaching institute so you might not be able to find it but it is the FIITJEE JEE Adv Review Package Part II. – Gurasees May 04 '23 at 16:26
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Acha okay, Ill post my answer as soon as i get it – NadiKeUssPar May 04 '23 at 16:32
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Does this answer your question? Find $\frac{\sum_\limits{k=0}^{6}\csc^2\left(x+\frac{k\pi}{7}\right)}{7\csc^2(7x)}$ - found using an Approach0 search. There are also other duplicates, e.g., ... – John Omielan May 04 '23 at 16:40
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(cont.) Evaluating $\frac{\sum_{k=0}^6 \csc^2(a+\frac{k\pi}{7})}{7\csc^2(7a)}$, Value of $\frac{\sum_{k=0}^{6} \csc^{2}(\theta + \frac{k\pi}{7})}{7\csc^{2}(7\theta)}$, etc. Last, but not least, welcome to Math SE. – John Omielan May 04 '23 at 16:42
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1oh yes it does help, and thanks. I have been helped by this community for a long time and now finally I am registered and will contribute!! – Gurasees May 04 '23 at 16:45