Additivity of arctan &#x2061;<!-- ⁡ --> ( x + y </mrow

CorRoorgofmq2b

CorRoorgofmq2b

Answered question

2022-06-01

Additivity of arctan ( x + y 1 x y )

Answer & Explanation

Rhett Pruitt

Rhett Pruitt

Beginner2022-06-02Added 5 answers

Differentiating,
d d x arctan ( x + y 1 x y ) = 1 1 + ( x + y ) 2 / ( 1 x y ) 2 ( 1 1 x y ( x + y ) y ( 1 x y ) 2 ) = 1 x y + x y + y 2 ( 1 x y ) 2 + ( x + y ) 2 = 1 + y 2 1 2 x y + x 2 y 2 + x 2 + 2 x y + y 2 = 1 + y 2 1 + x 2 + y 2 + x 2 y 2 = 1 + y 2 ( 1 + y 2 ) ( 1 + x 2 ) = 1 1 + x 2 ,
and then integrating gives
arctan ( x + y 1 x y ) = arctan x + C ( y ) .
Putting x=0 gives C ( y ) = arctan y

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