Need to evaluate lim n → + ∞ ∏ r = 1 n [ 1 − tan 2 ⁡ θ 2 r ] . I can't get the given answer, which is θ tan ⁡ θ

Question

Need to evaluate       lim          n      →      +      ∞            ∏          r      =      1        n        [    1    −          tan      2        ⁡          θ              2        r              ]  . I can&#039;t get the given answer, which is       θ          tan      ⁡      θ

postotnojeyf · Accepted Answer

It has a much simpler solution:note that   1  −      tan    2    ⁡  (  x  )  =            2      tan      ⁡      (      x      )              tan      ⁡      (      2      x      )      , hence we have that      ∏          r      =      1              n        (  1  −      tan    2    ⁡  (      θ          2      r        )  )  =      ∏          r      =      1              n                  2      tan      ⁡      (              θ                  2          r                    )              tan      ⁡      (              θ                  2                      r            −            1                              )        =                    2        n            tan      ⁡      (              θ                  2          n                    )              tan      ⁡      (      θ      )      and now it is easy to see that       lim          n      →      ∞            2    n    tan  ⁡  (      θ          2      n        )  =      lim          n      →      ∞            2    n    ⋅      θ          2      n        =  θ as desired.

Kenna Stanton · Accepted Answer

I think your calculate for       ∏          r      =      1        n    cos  ⁡      (          θ              2        r              )   is wrong. In fact,                                          ∏                      r            =            1                    n                cos        ⁡                  (                      θ                          2              r                                )                                                  =                    cos        ⁡                  (                      θ            2                    )                cos        ⁡                  (                      θ                          2              2                                )                ⋯        cos        ⁡                  (                      θ                          2              n                                )                                                  =                                          cos            ⁡                          (                              θ                2                            )                        cos            ⁡                          (                              θ                                  2                  2                                            )                        ⋯            cos            ⁡                          (                              θ                                  2                                      n                    −                    1                                                              )                        ⋅            2            cos            ⁡                          (                              θ                                  2                  n                                            )                        sin            ⁡                          (                              θ                                  2                  n                                            )                                            2            sin            ⁡                          (                              θ                                  2                  n                                            )                                                                        =                                          cos            ⁡                          (                              θ                2                            )                        cos            ⁡                          (                              θ                                  2                  2                                            )                        ⋯            cos            ⁡                          (                              θ                                  2                                      n                    −                    1                                                              )                        ⋅            sin            ⁡                          (                              θ                                  2                                      n                    −                    1                                                              )                                            2            sin            ⁡                          (                              θ                                  2                  n                                            )                                                                        =                                          cos            ⁡                          (                              θ                2                            )                        cos            ⁡                          (                              θ                                  2                  2                                            )                        ⋯            cos            ⁡                          (                              θ                                  2                                      n                    −                    2                                                              )                        ⋅            sin            ⁡                          (                              θ                                  2                                      n                    −                    2                                                              )                                                          2              2                        sin            ⁡                          (                              θ                                  2                  n                                            )                                                                        =                    ⋯                                          =                                          cos            ⁡                          (                              θ                2                            )                        sin            ⁡                          (                              θ                2                            )                                                          2                              n                −                1                                      sin            ⁡                          (                              θ                                  2                  n                                            )                                                                        =                                          cos            ⁡                          (              θ              )                                                          2                              n                                      sin            ⁡                          (                              θ                                  2                  n                                            )                                      .

Need to evaluate limn→+∞∏r=1n[1−tan2θ2r].

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