I know g ( x ) = arctan ⁡ ( x ) + arctan ⁡...

ban1ka1u

ban1ka1u

Answered

2022-07-07

I know
g ( x ) = arctan ( x ) + arctan ( y ) = arctan ( x + y 1 x y )

Answer & Explanation

torpa6d

torpa6d

Expert

2022-07-08Added 7 answers

Fix, as usual:
π 2 < γ = arctan ( t ) < π 2
now we have:
tan ( γ ) = tan ( α + β ) = x + y 1 x y = t
and, if x y > 1 we have the two cases (x and y have the same sign):
x > 0 , y > 0 t < 0 γ < 0 α + β = γ + π
x < 0 , y < 0 t > 0 γ > 0 α + β = γ π
Lucia Grimes

Lucia Grimes

Expert

2022-07-09Added 5 answers

I can prove that if | x y | < 1, that
1)
π 2 < arctan ( x ) + arctan ( y ) < π 2
2)
arctan ( x ) + arctan ( y ) = arctan ( x + y 1 x y )

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