\[\large \sum_{r=1}^{n} \dfrac{1-\cos\left(\frac{(2r-1)\pi}{2n}\right)} {\left[1+\cos\left(\frac{(2r-1)\pi}{2n}\right)\right]\left[5+3\cos\left(\frac{(2r-1)\pi}{2n}\right)\right]^2} = \dfrac{n(8n-11)}{16} \]

Prove that for integer \(n>1\), the equation above holds true.

Please try not to use induction (assuming it's possible).

This problem was copied from another math forum but no one responded to it.

This is a part of the set Formidable Series and Integrals.

## Comments

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TopNewest@Aditya Kumar bache ki jaan lo ge kya thoda to hint de the Bhaiya. – Lakshya Sinha · 1 year, 1 month ago

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have you tried roots of unity – Aareyan Manzoor · 1 year, 1 month ago

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– Pi Han Goh · 1 year, 1 month ago

How does that help? We can only use that if we can split out all the expressions and calculate them separately. \( \sum \dfrac f{g \cdot h} \ne \sum f \cdot \sum \dfrac1g \cdot \sum \dfrac1h \).Log in to reply

– Aareyan Manzoor · 1 year, 1 month ago

I havent tried this yet but we could use partial fractions and then f'/f or maybe chebyshev polynomials of the first kind.....Log in to reply