Regarding an inverse trigonometric equation.I tried to find the solutions of this equation arctan ⁡ ( 2 x 1 − x 2 ) + arccot ( 1 − x 2 2 x ) = 2 π 3 I got solutions 1 3 and − 3 . by reciprocating the arc cot term into arc tan term, adding both and solving the equation. But in solutions, in addition to above answers, 3 + 2 and 3 − 2 has also been given as answers, which I cannot figure out how they came?

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Regarding an inverse trigonometric equation.I tried to find the solutions of this equation  arctan  ⁡      (                  2        x                    1        −                  x          2                      )    +  arccot      (                  1        −                  x          2                            2        x              )    =            2      π        3  I got solutions       1          3       and   −      3  . by reciprocating the arc cot term into arc tan term, adding both and solving the equation. But in solutions, in addition to above answers,       3    +  2 and       3    −  2 has also been given as answers, which I cannot figure out how they came?

Savion Chaney · Accepted Answer

Please note that:  arctan  ⁡      (                  2        x                    1        −                  x          2                      )    +  arccot      (                  1        −                  x          2                            2        x              )  =      {                            2          arctan          ⁡                      (                                          2                x                                            1                −                                  x                  2                                                      )                    ,                                      if                                           (                                                      2                    x                                                        1                    −                                          x                      2                                                                      )                                                    &amp;gt;                                      0                                                            2          arctan          ⁡                      (                                          2                x                                            1                −                                  x                  2                                                      )                    +          π          ,                                      if                                           (                                                      2                    x                                                        1                    −                                          x                      2                                                                      )                                                                 &amp;lt;                                      0                                              . In the first case, you get x=1/      3  ,x=-      3  . In the second case, you get x=      3  +2,      3  -2.P.S. I am using principal values for all inverse trigonometric ratios involved in here.

Aliyah Thompson · Accepted Answer

Use:Let       cot          −      1        ⁡  x  =  θ  x  =  cot  ⁡  θ                                          tan                          −              1                                ⁡                      1            x                    =          θ                                                              tan                          −              1                                ⁡          x          +                      tan                          −              1                                ⁡          y          =                      tan                          −              1                                ⁡                                    x              +              y                                      1              −              x              y                                                  (Not necessarily used)So      cot          −      1        ⁡            1      −              x        2                    2      x        =      tan          −      1        ⁡            2      x              1      −              x        2                  tan          −      1        ⁡            2      x              1      −              x        2              =      π    3              2      x              1      −              x        2              =      3    2  x  =      3    −      3        x    2

Regarding an inverse trigonometric equation. I tried to find the solutions of this equation arctan ((2x)/(1-x^2))+ arccot((1-x^2)/(2x))=(2 pi)/(3)

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