Inverse trigonometric equation tan − 1 ⁡ x + cot − 1 ⁡ x = π 2 I think there have to be used formula: tan ⁡ ( α + β ) = tan ⁡ α + tan ⁡ β 1 − tan ⁡ α tan ⁡ β but I don't know how to apply it and yet I don't know whether it's true that cot − 1 ⁡ x = 1 tan − 1 ⁡ x ?

Question

Inverse trigonometric equation       tan          −      1        ⁡      x    +      cot          −      1        ⁡      x    =      π    2   I think there have to be used formula:       tan    ⁡    (    α    +    β    )    =                  tan        ⁡                  α                +        tan        ⁡                  β                            1        −        tan        ⁡                  α                tan        ⁡                  β                     but I don&#039;t know how to apply it and yet I don&#039;t know whether it&#039;s true that             cot              −        1              ⁡          x        =          1                        tan                      −            1                          ⁡                  x                    ?

Deanna Sweeney · Accepted Answer

Let   y  =  arctan  ⁡  x, then                    tan        ⁡        y                            =        x                                      1                      tan            ⁡            y                                              =                  1          x                                    cot        ⁡        y                            =                  1          x                                    tan        ⁡                  (                      π            2                    −          y          )                                    =                  1          x                                              π          2                −        y                            =        arctan        ⁡                  (                      1            x                    )                                              π          2                −        arctan        ⁡        x                            =        arccot                 x                                                  arctan            ⁡            x            +            arccot                         x                                                =                      π            2                          .                        ◼            Here is the link for the proof of      arctan  ⁡      (          1      x        )    =  arccot  (  x  )

inverse trigonometric equation tan^(-1)(x)+cot^(-1)(x)=(pi)/(2)

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