Show that 4 cos ⁡ π 5 ⋅ sin ⁡ π 10 = 1

Question

Show that   4  cos  ⁡      π    5    ⋅  sin  ⁡      π    10    =  1

redclick53 · Accepted Answer

Using Werner Formula,  2  cos  ⁡      36    ∘    sin  ⁡      18    ∘    =  sin  ⁡      54    ∘    −  sin  ⁡      18    ∘    =  cos  ⁡      36    ∘    −  cos  ⁡      72    ∘  Now see Proving trigonometric equation   cos  ⁡  (      36    ∘    )  −  cos  ⁡  (      72    ∘    )  =  1      /    2

Rachel Villa · Accepted Answer

Given       2    cos    ⁡          π      5        ⋅    sin    ⁡          π      10            ,  , Now Let             π      10        =    θ        ,  , and using  2  sin  ⁡  A  ⋅  cos  ⁡  A  =  sin  ⁡  2  AThen      2    cos    ⁡    2    θ    ⋅    sin    ⁡    θ    =                  2        cos        ⁡        2        θ        ⋅        sin        ⁡        θ        ⋅        cos        ⁡        θ                    cos        ⁡        θ              =                  2        sin        ⁡        2        θ        ⋅        cos        ⁡        2        θ                    2        cos        ⁡        θ              =                  sin        ⁡        4        θ                    2        cos        ⁡        θ            So we get                    sin        ⁡        4        θ                    2        cos        ⁡        θ              =                  sin        ⁡                  (                                    4              π                        10                    )                            2        cos        ⁡                  π          10                      =                  cos        ⁡                  π          10                            2        cos        ⁡                  π          10                      =          1      2      Bcz      sin    ⁡                  4        π            10        =    sin    ⁡          (              π        2            −              π        10            )        =    cos    ⁡          π      10

Show that 4 cos ⁡ π 5 </mfrac> &#x22

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