Prove cos ⁡ x + cos ⁡ ( 2 x ) + ⋯ + cos ⁡ ( n x ) = sin ⁡ ( n x 2 ) cos ⁡ ( n + 1 ) x 2 sin ⁡ ( x 2 ) . ( 1 )

Question

Prove   cos  ⁡  x  +  cos  ⁡  (  2  x  )  +  ⋯  +  cos  ⁡  (  n  x  )  =            sin      ⁡      (                        n          x                2            )      cos      ⁡                        (          n          +          1          )          x                2                    sin      ⁡      (              x        2            )        .     (  1  )

Kaelyn Barrett · Accepted Answer

Here is the induction step: it comes down to proving                                          sin            ⁡                                                            n                  x                                2                                      cos            ⁡                                                            (                  n                  +                  1                  )                  x                                2                                                          sin            ⁡                                          x                2                                                    +        cos        ⁡        (        n        +        1        )        x        =                              sin            ⁡                                                            (                  n                  +                  1                  )                  x                                2                                      cos            ⁡                                                            (                  n                  +                  2                  )                  x                                2                                                          sin            ⁡            (                                          x                2                                      )                                              or                sin        ⁡                                            n              x                        2                          cos        ⁡                                            (              n              +              1              )              x                        2                          +        sin        ⁡                              x            2                          cos        ⁡        (        n        +        1        )        x        =        sin        ⁡                                            (              n              +              1              )              x                        2                          cos        ⁡                                            (              n              +              2              )              x                        2                              Now use the linearisation formulae:      {                                        sin            ⁡                                          n                x                            2                        cos            ⁡                                          (                n                +                1                )                x                            2                        =                                          1                2                                                                    (                                      sin            ⁡                                          (                2                n                +                1                )                x                            2                        −            sin            ⁡                          x              2                                                      )                                      ,                                                sin          ⁡                                    x              2                                cos          ⁡          (          n          +          1          )          x          =                      1            2                                              (                                sin          ⁡                                                    (                2                n                +                3                )                x                            2                                −          sin          ⁡                                                    (                2                n                +                1                )                x                            2                                                          )                                ,                        whence the l.h.s. is      1    2              (        sin  ⁡                    (        2        n        +        3        )        x            2        −  sin  ⁡      x    2              )        =  sin  ⁡            (      n      +      1      )      x        2      cos  ⁡            (      n      +      2      )      x        2  by the factorisation formula:     sin  ⁡  p  −  sin  ⁡  q  =  2  sin  ⁡                    p        −        q            2        cos  ⁡                    p        +        q            2

Prove cos ⁡ x + cos ⁡ ( 2 x ) + ⋯<

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