How to find the exact value of sec 75 using the half angle formula?

Question

How to find the exact value of             sec      75       using the half angle formula?

Barrecajhoe · Accepted Answer

As             sec      75        =          1              cos        75                    , we have to first find the value of                     cos      75      .The following Half-Angle Formula must be applied:             cos              (                  θ          2                )              =    ±                                        1            +                          cos              θ                                2                      ,   where, the sign (+ or -) is to be determined in accordance with             cos              (                  θ          2                )            Taking,             θ      2        =    75    ,    i    .    e    .    ,    θ    =    150  , and, noting that       75   lies in the First Quadrant , where,             cos                       is                       +    v    e  , we get,            cos      75        =    +                                        1            +                          cos              150                                2                      =                            (          1          +                                    cos                              (                180                -                30                )                                      2                    )                          =                                        1            -                          cos              30                                2                      =                                        1            -                                          3                            2                                2                      =                                        2            -                          3                                4                          =          1      2                      2        -                  3                      =          1      2                      2        -        2                                            3              4                                                =          1      2                                3          2                +                  1          2                -        2                                            3              2                        ⋅                          1              2                                                =          1      2                                                                          3                2                                              2                +                                                            1                2                                              2                -        2        ⋅                                            3              2                                      ⋅                                            1              2                                                =          1      2                                            (                                                            3                  2                                                      -                                                            1                  2                                                      )                    2                          =          1      2              (                                    3            2                              -                                    1            2                              )        =                            3                -        1                    2                  2                    .Hence,             sec      75        =          1              cos        75              =                  2                  2                                      3                -        1                  =                  2                  2                                      3                -        1              ×                            3                +        1                              3                +        1                  =          2              (              3            -      1      )            =          6        -          2      .

How to find the exact value of sec75 using the half angle formula?

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