How do you prove: 1 sin 2 ⁡ ( 2 π 9 ) − 1 sin 2 ⁡ ( 4 π 9 ) = 8 sin ⁡ ( π 18 )

Question

How do you prove:             1                        sin          2                ⁡        (                              2            π                    9                )              −            1                        sin          2                ⁡        (                              4            π                    9                )              =  8  sin  ⁡  (      π    18    )

siriceboynu1 · Accepted Answer

Let       c    n    :=  cos  ⁡            n      π        9    ,        s    n    :=  sin  ⁡            n      π        9   so we want to prove  α  :=            s      6                      s        2                    s        4        2              =  8      c    4    .Since       s          9      −      n        =      s    n   and       s    3    =            3        2        α          8              c        4              =            3              8              s        1                    s        2                    s        4              .Now we just need to prove      s    1        s    2        s    4    =            3        8    ,which is a duplicate.

How do you prove: 1/(sin^2((2 pi)/(9)))-1/(sin^2((4pi)/9))=8 sin (pi/18)

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