Prove that cos ⁡ ( 2 n x ) = ∑ k = 0 n ( − 1 ) k ( 2 n 2 k ) cos 2 ( n − k ) ⁡ ( x ) ⋅ sin 2 k ⁡ ( x ) := p ( n )

Question

Prove that   cos  ⁡  (  2  n  x  )  =      ∑          k      =      0        n    (  −  1      )    k                                (                                      2          n                          2          k                                      )                          cos          2      (      n      −      k      )        ⁡  (  x  )  ⋅      sin          2      k        ⁡  (  x  )  :=  p  (  n  )

Calvin Oneill · Accepted Answer

Not by induction:                    cos        ⁡        2        n        x                            =        ℜ        [        cos        ⁡        2        n        x        +        i        sin        ⁡        2        n        x        ]                                          =        ℜ        [        (        cos        ⁡        x        +        i        sin        ⁡        x                  )                      2            n                          ]                                          =        ℜ                  [                      ∑                          j              =              0                                      2              n                                                                          (                                                                    2                n                            j                                                      )                                                          cos                          2              n              −              j                                ⁡          x          ⋅                      i            j                                sin            j                    ⁡          x          ]                                                  =                  ∑                      k            =            0                                n                                                              (                                                          2              n                                      2              k                                                          )                                                cos                      2            n            −            2            k                          ⁡        x        ⋅                  i                      2            k                                    sin                      2            k                          ⁡        x                                  (        j        =        2        k        )                                          =                  ∑                      k            =            0                                n                          (        −        1                  )                      k                                                              (                                                          2              n                                      2              k                                                          )                                                cos                      2            (            n            −            k            )                          ⁡        x        ⋅                  sin                      2            k                          ⁡        x

Prove that cos ⁡ ( 2 n x ) = <msubsup> ∑

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