How do i show this equality without using reccurence method: ∏ k = 0 k = n cos ⁡ θ 2 k = sin ⁡ θ 2 n sin ⁡ ( 2 − n θ ) ?

Question

How do i show this equality without using reccurence method:       ∏          k      =      0              k      =      n            cos    ⁡          θ              2        k              =                    sin        ⁡        θ                              2          n                sin        ⁡        (                  2                      −            n                          θ        )            ?

hoffwnbu · Accepted Answer

First, notice that:                    2        cos        ⁡        x        sin        ⁡        x                            =        sin        ⁡        2        x                                    ⟹                cos        ⁡        x                            =                  1          2                                      sin            ⁡            2            x                                sin            ⁡            x                              Then, given an integer number n&amp;gt;0 we have  cos  ⁡      (          θ      2        )    ⋅  cos  ⁡      (          θ              2        2              )    ⋅  …  ⋅  cos  ⁡      (          θ              2        n              )    =      1          2      n                  sin      ⁡              (        θ        )                    sin      ⁡              (                  θ          2                )              ⋅            sin      ⁡              (                  θ          2                )                    sin      ⁡              (                  θ                      2            2                          )              ⋅  …  ⋅            sin      ⁡              (                  θ                      2                          n              −              1                                      )                    sin      ⁡              (                  θ                      2            n                          )            Which is a &quot;telescopic product&quot;, and the inequality follows from here.

How do i show this equality without using reccurence method: <munderover> ∏<!-- ∏ --

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