How to prove sinh − 1 ⁡ ( tan ⁡ x ) = log ⁡ tan ⁡ ( π 4 + x 2 )I have tried using many identity but in vainFor reference tanh − 1 ⁡ x = 1 2 log ⁡ 1 + x 1 − x and sinh − 1 ⁡ x = log ⁡ ( x + x 2 + 1 )

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How to prove       sinh          −      1        ⁡  (  tan  ⁡  x  )  =  log  ⁡  tan  ⁡  (      π    4    +      x    2    )I have tried using many identity but in vainFor reference      tanh          −      1        ⁡  x  =      1    2    log  ⁡            1      +      x              1      −      x      and      sinh          −      1        ⁡  x  =  log  ⁡  (  x  +            x      2        +    1    )

Arcatuert3u · Accepted Answer

Putting tanx in place of x in this formula      sinh          −      1        ⁡  x  =  log  ⁡  (  x  +            x      2        +    1    )we have,      sinh          −      1        ⁡  tan  ⁡  x  =  log  ⁡  (  tan  ⁡  x  +      (    tan    ⁡    x          )      2        +    1    )  =  log  ⁡  (  tan  ⁡  x  +      (    sec    ⁡    x          )      2        )  =  log  ⁡  (  tan  ⁡  x  +  sec  ⁡  x  )  =  log  ⁡  (            1      +      sin      ⁡      x              cos      ⁡      x        )  =  log  ⁡      [                  (        sin        ⁡                  x          2                +        cos        ⁡                  x          2                          )          2                                      cos          2                ⁡                  x          2                −                  sin          2                ⁡                  x          2                      ]    =  log  ⁡      [                  sin        ⁡                  x          2                +        cos        ⁡                  x          2                            cos        ⁡                  x          2                −        sin        ⁡                  x          2                      ]    =  log  ⁡      [                  1        +        tan        ⁡                  x          2                            1        −        tan        ⁡                  x          2                      ]    =  log  ⁡      [                  tan        ⁡                  π          4                +        tan        ⁡                  x          2                            1        −        tan        ⁡                  π          4                tan        ⁡                  x          2                      ]    =  log  ⁡  tan  ⁡  (      π    4    +      x    2    )Hence proved.

Emanuel Keith · Accepted Answer

Notice,      sinh          −      1        ⁡  (  tan  ⁡  x  )  =  log  ⁡  (  tan  ⁡  x  +            tan      2        ⁡    x    +    1    )  =  log  ⁡  (  tan  ⁡  x  +  sec  ⁡  x  )  =  log  ⁡      (    tan    ⁡    x    +          1              cos        ⁡        x              )    =  log  ⁡      (                  2        tan        ⁡                  x          2                            1        −                  tan          2                ⁡                  x          2                      +                  1        +                  tan          2                ⁡                  x          2                            1        −                  tan          2                ⁡                  x          2                      )    =  log  ⁡      (                            (          1          +          tan          ⁡                      x            2                    )                2                              (          1          −          tan          ⁡                      x            2                    )                          (          1          +          tan          ⁡                      x            2                    )                      )    =  log  ⁡      (                  1        +        tan        ⁡                  x          2                            1        −        tan        ⁡                  x          2                      )    =  log  ⁡      (                  tan        ⁡                  π          4                +        tan        ⁡                  x          2                            1        −        tan        ⁡                  π          4                tan        ⁡                  x          2                      )    =      log    ⁡    tan    ⁡          (              π        4            +              x        2            )

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