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Wesley Hicks 2022-06-02 Answered
Proving
n = 1 , 3 , 5.. 4 k sin 2 ( n k ) n 2 = π
Where k any number greater than 0
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Answers (1)

a1p2ung1ls6t7
Answered 2022-06-03 Author has 6 answers
Using (in the interval ( 0 , π ) )
2 π x x 2 8 = n 1 sin ( n x 2 ) 2 n 2 ( 1 )
we have
2 π x x 2 8 = n 1 sin ( ( 2 n 1 ) x 2 ) 2 ( 2 n 1 ) 2 + 1 4 n 1 sin ( n x ) 2 n 2
and using (1) again we get
1 4 n 1 sin ( n x ) 2 n 2 = π x x 2 8
so
π x 8 = n 1 sin ( ( 2 n 1 ) x 2 ) 2 ( 2 n 1 ) 2
now put
x = 2 k
and we have
π 4 k = n 1 sin ( ( 2 n 1 ) 1 k ) 2 ( 2 n 1 ) 2
as wanted.
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