Show that, 2 arctan ⁡ ( 1 3 ) + 2 arcsin ⁡ ( 1 5 2 ) − arctan ⁡ ( 1 7 ) = π 4 There is a mixed of sin and tan, how can I simplify this to π 4 We know the identity of arctan ⁡ ( 1 a ) + arctan ⁡ ( 1 b ) = arctan ⁡ ( a + b a b − 1 )

Question

Show that,  2  arctan  ⁡      (          1      3        )    +  2  arcsin  ⁡      (          1              5                  2                      )    −  arctan  ⁡      (          1      7        )    =      π    4  There is a mixed of sin and tan, how can I simplify this to       π    4  We know the identity of   arctan  ⁡      (          1      a        )    +  arctan  ⁡      (          1      b        )    =  arctan  ⁡      (                  a        +        b                    a        b        −        1              )

Dwayne James · Accepted Answer

arctan  ⁡  x  +  arctan  ⁡  y  =      {                            arctan          ⁡                                                    x                +                y                                            1                −                x                y                                                                          if                     x          y          &amp;lt;          1                                      π          +          arctan          ⁡                                                    x                +                y                                            1                −                x                y                                                                          if                     x          y          &amp;gt;          1                            ⟹    2  arctan  ⁡  x  =  arctan  ⁡            2      x              1      −              x        2               if       x    2    &amp;lt;  1Finally use  arcsin  ⁡  x  =  arctan  ⁡            x              1        −                  x          2

gvaldytist · Accepted Answer

It would be easier to attack if you would substitute       tan          −      1        ⁡            1      7         for       sin          −      1        ⁡            1              5                  2

Show that, 2 arctan ⁡ ( 1 3 </mfrac>

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