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Sparse Sine Settlement (Posted on 2014-09-03) Difficulty: 3 of 5
Find the minimum positive integer N such that:

(sin 45o)-1*(sin 46o)-1+ (sin 47o)-1*(sin 48o)-1+.....+ (sin 133o)-1*(sin 134o)-1
= (sin No)-1

No Solution Yet Submitted by K Sengupta    
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Some Thoughts An identity to prove Comment 2 of 2 |
(sin x)^-1 is just csc x, so I'll write everything in terms of csc.

Add in the 'skipped' terms (csc 46)*(csc 47) + (csc 48)*(csc 49) + ... +(csc 134)*(csc 135).  These are duplicate values of the terms given (but in reverse order), so the sum of the series with these extra terms is 2*csc(N).

Based on the rather convenient answer of N=1, I suspect this is a specific instance of an identity:

For any positive integer n: 
2*csc(90/n) = sum{x=1 to n} csc(45+(x-1)*90/n)*csc(45+x*90/n)

Edited on June 24, 2017, 12:45 pm
  Posted by Brian Smith on 2017-06-24 12:44:20

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