The value of tan − 1 ⁡ ( 1 2 ) − tan − 1 ⁡ ( 5 − 2 6 1 + 6 ) is equal to π 6 π 4 π 3 π 12

Question

The value of       tan          −      1        ⁡      (          1              2              )    −      tan          −      1        ⁡      (                  5        −        2                              6                                      1        +                  6                      )  is equal to      π    6        π    4        π    3        π    12

s23160125909p46 · Accepted Answer

5        −        2                  6                            1        +                  6                      =                              3                −                  2                            1        +                  3                ⋅                  2                        ⟹    arctan  ⁡                    5        −        2                  6                            1        +                  6                      =  arctan  ⁡      3    −  arctan  ⁡      2    arctan  ⁡      3    =            π      3      and  arctan  ⁡      2    =  arccot            1              2              =            π      2        −  arctan  ⁡            1              2

raukie1v3yr · Accepted Answer

By using identities      5    −    2          6        =      3    −      2  and      tan          −      1        ⁡      α    −      tan          −      1        ⁡      β    =      tan          −      1        ⁡  (            α      −      β              1      +      α      β        )  ,after a little calculation, we get      tan          −      1        ⁡      (          1              2              )    −      tan          −      1        ⁡      (                  5        −        2                              6                                      1        +                  6                      )    =      tan          −      1        ⁡  (      1          3        )  =      π    6

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