How can I prove that : 2 arctan ⁡ ( x ) + arcsin ⁡ ( 2 x 1 + x 2 ) = π , x&gt;1What is the best way to do this ?

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How can I prove that :  2  arctan  ⁡  (  x  )  +  arcsin  ⁡            (                  2      x              1      +              x        2                        )        =  π , x&amp;gt;1What is the best way to do this ?

upornompe · Accepted Answer

An easy way is to differentiate the function  f  (  x  )  =  2  arctan  ⁡  (  x  )  +  arcsin  ⁡      (                            2          x                          1          +                      x            2                                )  and show that f′(x)=0 so that f is a constant. Then calculate the constant by letting x be a well-chosen number.

deceptie3j · Accepted Answer

You can prove it without differentiating: set   t  =  2  arctan  ⁡  x. This means  tan  ⁡      t    2    =  x    and    −  π  &amp;lt;  t  &amp;lt;  π  .Furthermore, as x&amp;gt;1, we have               π      2        &amp;lt;  t  &amp;lt;  πNow, by the half-angle formulae (actually this is where the formula we seek to prove comes from),  sin  ⁡  t  =            2      x              1      +              x        2              ,    whence    t  ≡      {                            arcsin          ⁡                                    2              x                                      1              +                              x                2                                                        or                                      π          −          arcsin          ⁡                                    2              x                                      1              +                              x                2                                                                mod      2  π  .Knowing               π      2        &amp;lt;  t  &amp;lt;  π, we necessarily have  t  =  2  arctan  ⁡  x  =  π  −  arcsin  ⁡            2      x              1      +              x        2              .

How can I prove that : 2 arctan ⁡ ( x ) + arcsin &#x2

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