Indeed the saying "Curiosity is the mother of inventions" is absolutely true. While learning trigonometry , I found this:

Prove that in \(\Delta ABC\) ,

\[\large\sum_{cyc} \cot A = \prod_{cyc} \csc A + \prod_{cyc} \cot A\]

I invite all brilliantians to prove this :)

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TopNewestSince, \(A+B+C = 180^{o}\)

\[Cos(A+B+C) = -1\]

\[ \prod_{cyc} \cos (A) - \sum_{cyc} \cos (A) \sin (B) \sin (C) = -1\]

Divide both sides with \(\prod_{cyc} \sin (A)\),

\[\prod_{cyc} \cot (A) - \sum_{cyc} \cot (A) = - \prod_{cyc} \csc (A)\]

\[\sum_{cyc} \cot (A) = \prod_{cyc} \cot (A) + \prod_{cyc} \csc (A)\]

Hence, proved. – Surya Prakash · 1 year, 5 months ago

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@Nihar Mahajan How is my solution? Is there any other method u r thinking? – Surya Prakash · 1 year, 5 months ago

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– Nihar Mahajan · 1 year, 5 months ago

It is quite shorter and elegant than mine.Nice work!Log in to reply

– Sudeep Salgia · 1 year, 5 months ago

Wait isn't that what I had told you?Log in to reply

– Nihar Mahajan · 1 year, 5 months ago

That day , we both forgot that \(\cos(A+B+C)=-1\). Don't worry , it happens sometimes :PLog in to reply

– Surya Prakash · 1 year, 5 months ago

Do u have another method? Can u please post it?Log in to reply

– Nihar Mahajan · 1 year, 5 months ago

I have another method , but it does not use basic results. It uses "lemma type" results , thats because I found this equation while proving that lemma only!Log in to reply

– Surya Prakash · 1 year, 5 months ago

Please send that solution . I am eager to know another solution to it. PleaseLog in to reply

Hello Nihar! Can you please tell me from where are you learning trigonometry?? Thanks 😀 – Harsh Shrivastava · 1 year, 5 months ago

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– Nihar Mahajan · 1 year, 5 months ago

M Prakash Academy , Pune.Log in to reply

Exactly what is "\(cyc\)" in \(\displaystyle \sum_{cyc}\) and \(\displaystyle\prod_{cyc}\)? \(~~~\) Never seen this. – Micah Wood · 1 year, 5 months ago

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It is short form of the cyclic summation/product.I hope you get it now :) – Nihar Mahajan · 1 year, 5 months ago

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– Dev Sharma · 1 year, 4 months ago

what does sym summation mean?Log in to reply