Show that sin ⁡ ( m x ) sin ⁡ ( x ) = ∑ j = 0 , j ≡ 1 ( mod 2 ) m ( m j ) ( 1 − sin 2 ⁡ ( x ) ) ( m − j ) / 2 ( − sin 2 ⁡ ( x ) ) ( j − 1 ) / 2 where m is odd. I did as:If z = e i x , then, sin ⁡ ( m x ) sin ⁡ ( x ) = ℑ ( z m ) ℑ ( z ) = z m − z ¯ m z − z ¯ = ∏ k = 1 m ( z − ζ m k z ¯ ) ,where ζ m is a primitive root of X m − 1Any idea on how to conclude ?

Question

Show that            sin      ⁡      (      m      x      )              sin      ⁡      (      x      )        =      ∑          j      =      0      ,      j      ≡      1            (      mod            2      )        m                      (                    m      j                      )              (  1  −      sin    2    ⁡  (  x  )      )          (      m      −      j      )              /            2        (  −      sin    2    ⁡  (  x  )      )          (      j      −      1      )              /            2      where m is odd. I did as:If   z  =      e          i      x      , then,            sin      ⁡      (      m      x      )              sin      ⁡      (      x      )        =            ℑ      (              z        m            )              ℑ      (      z      )        =                    z        m            −                                    z            ¯                          m                    z      −                        z          ¯                      =      ∏          k      =      1        m    (  z  −      ζ    m    k              z      ¯        )  ,where       ζ    m   is a primitive root of       X    m    −  1Any idea on how to conclude ?

fongama33 · Accepted Answer

Hint                    sin        ⁡        (        m        x        )                            =        Im        ⁡        (                  e                      i            x            m                          )                                          =                              (                          e                              i                x                                                    )              m                        −            (                          e                              −                i                x                                                    )              m                                            2            i                                                            =                              (            cos            ⁡            x            +            i            sin            ⁡            x                          )              m                        −            (            cos            ⁡            x            −            i            sin            ⁡            x                          )              m                                            2            i                                                            =        −        i                  ∑                                                                      odd                                                                                j                                                                              (                                                    m              j                                                      )                                                    (        cos        ⁡        x                  )                      m            −            j                          (        i        sin        ⁡        x                  )          j                                                  =        −        i                  ∑                                                                      odd                                                                                j                                                                              (                                                    m              j                                                      )                                                    (        1        −                  sin          x                          )                                                    m                −                j                            2                                      (        −                  sin          2                ⁡        x                  )                                    j              2

Show that sin ⁡ ( m x ) </mrow>

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