How to use a half-angle formula to simplify tan 105 ?

Question

How to use a half-angle formula to simplify             tan      105      ?

Lekno2y5t · Accepted Answer

A derivation for             tan              (                  x          2                )             is at the bottom.            tan              (                  x          2                )              =    ±                                        1            -                          cos              x                                            1            +                          cos              x                                          depending on the quadrant of             105      o       (quadrant II, thus the horizontal axis       &amp;lt;    0   on the unit circle).                              tan          105                o            =              tan              (                              7            π                    12                )              =          tan              (                  1          2                ⋅                              7            π                    6                )              =    -                                        1            -                          cos                              (                                                      7                    π                                    6                                )                                                          1            +                          cos                              (                                                      7                    π                                    6                                )                                                              =    -                                        1            +                                          3                            2                                            1            -                                          3                            2                                                =    -                                                                      2                +                                  3                                            2                                                                          2                -                                  3                                            2                                                =    -                                        2            +                          3                                            2            -                          3                                                =    -                                                      (              2              +                              3                            )                        2                                              (              2              -                              3                            )                                      (              2              +                              3                            )                                                =    -                                                      (              2              +                              3                            )                        2                                      1            And since we already specified the quadrant, there&#039;s no need for       ±   (and       2    +          3        &amp;gt;    0   of course).      =    -          (      2      +              3            )            =    -    2    -          3      If you can&#039;t recall the half-angle formula, you can derive it.                    sin        2                    (        x        )              =                  1        -                  cos                      (            2            x            )                              2      Similarly:                    sin        2                    (                  x          2                )              =                  1        -                  cos                      (            x            )                              2      Thus:            |              sin                  (                      x            2                    )                    |        =                                        1            -                          cos              x                                2                    Similarly:                    cos        2                    (        x        )              =                  1        +                  cos                      (            2            x            )                              2                          cos        2                    (                  x          2                )              =                  1        +                  cos          x                    2                  |              cos                  (                      x            2                    )                    |        =                                        1            +                          cos              x                                2                    Now we can divide them.            |                        sin                      (                          x              2                        )                                                cos                          (                              x                2                            )                                          |        =          |              tan                  (                      x            2                    )                    |        =                                                      1              -                              cos                x                                      2                                                                          1              +                              cos                x                                      2                                    =                                        1            -                          cos              x                                            1            +                          cos              x

cumbresrugbyv3qs · Accepted Answer

tan 105 = tan (45 + 60)tan 45 = 1 ; tan 60 = sqrt3            tan      105        =                  1        +                  3                            1        -                  3

How to use a half-angle formula to simplify tan 105?

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