How can I simplify this complex number to get a real number? e i π a 2 [ 1 − e i π a ] [ 1 − e i 2 π a ] I am trying to arrive at 1 2 cos ⁡ ( π a 2 ) I've tried dividing top and bottom by one of the exponential terms, and also tried expanding out the Euler formulas for each exponential term to see if I get some cancellations and an obvious representation for cosine, but I have not been able to do itAny hints or solutions are welcome.Thanks!

Question

How can I simplify this complex number to get a real number?                              e                      i                                          π                a                            2                                      [        1        −                  e                      i            π            a                          ]                    [        1        −                  e                      i            2            π            a                          ]            I am trying to arrive at            1              2        cos        ⁡                  (                                    π              a                        2                    )                    I&#039;ve tried dividing top and bottom by one of the exponential terms, and also tried expanding out the Euler formulas for each exponential term to see if I get some cancellations and an obvious representation for cosine, but I have not been able to do itAny hints or solutions are welcome.Thanks!

verdesett014ci · Accepted Answer

Notice,                    e                  i                                    π              α                        2                              (      1      −              e                  i          π          α                    )              1      −              e                  i          2          π          α                      =                    e                                            i              π              α                        2                              (      1      −              e                  i          π          α                    )              1      −      (              e                  i          π          α                            )        2              =                    e                                            i              π              α                        2                              (      1      −              e                  i          π          α                    )              (      1      −              e                  i          π          α                    )      (      1      +              e                  i          π          α                    )        =            e                                    i            π            α                    2                            (      1      +              e                  i          π          α                    )        =      1                  e                                            −              i              π              α                        2                              (      1      +              e                  i          π          α                    )        =      1                  e                                            −              i              π              α                        2                              +              e                                            i              π              α                        2                                =      1          2              (                                            e                                                                    i                    π                    α                                    2                                                      +                          e                                                                    −                    i                    π                    α                                    2                                                              2                )              =      1          2      cos      ⁡              (                              π            α                    2                )

Daphne Fry · Accepted Answer

This involves a little judicious factoring of the numerator and denominator. so, here we go ...                                                        e                              i                π                a                                  /                                2                                      (            1            −                          e                              i                π                a                                      )                                1            −                          e                              i                2                π                a                                                                        =                                            e                              i                π                a                                  /                                2                                                    e                              i                π                a                                  /                                2                                      (                          e                              −                i                π                a                                  /                                2                                      −                          e                              i                π                a                                  /                                2                                      )                                              e                              i                π                a                                      (                          e                              −                i                π                a                                      −                          e                              i                π                a                                      )                                                                          =                              −            2            i            sin            ⁡            (            π            a                          /                        2            )                                −            2            i            sin            ⁡            (            π            a            )                                                                          =                              sin            ⁡            (            π            a                          /                        2            )                                2            sin            ⁡            (            π            a                          /                        2            )            cos            ⁡            (            π            a                          /                        2            )                                                                          =                  1                      2            cos            ⁡            (            π            a                          /                        2            )

How can I simplify this complex number to get a real number? <mstyle mathsize="1.2em">

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