Evaluating ∑ k = 0 6 csc 2 ⁡ ( a + k π 7...

Mylee Underwood

Mylee Underwood

Answered

2022-07-06

Evaluating k = 0 6 csc 2 ( a + k π 7 ) 7 csc 2 ( 7 a )

Answer & Explanation

treccinair

treccinair

Expert

2022-07-07Added 18 answers

The result of the summation is:
k = 0 6 csc 2 ( a + k π 7 ) 7 csc 2 ( 7 a ) = 7
The summation, after some calculation, is reduced to:
( A E F + B D F + C D E ) + D E F 7 D E F = 7
where
A = 4 2 2 + ( 2 2 2 ) c o s ( π 7 ),
B = 4 + 2 2 + ( 2 2 2 ) s i n ( π 14 )
C = 4 + 2 2 + ( 2 + 2 2 ) s i n ( 3 π 14 ),
D = 2 2 c o s ( π 7 ) + s i n ( 3 π 14 ) + 2,
E = c o s ( π 7 ) 2 2 s i n ( π 14 ) 2,
F = 2 s i n ( 3 π 14 ) + s i n ( π 14 ) 2.
By substituting and grouping we get:
6 + 1 8 7 1 8 = 7.

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