Given, for every x>1, f ( x ) = 4 arctan &#x2061;<!-- ⁡ --> 1 <m

Averi Mitchell

Averi Mitchell

Answered question

2022-06-13

Given, for every x>1,
f ( x ) = 4 arctan 1 x 1 + x
Show that f ( x ) = π 2 arctan ( x 1 )

Answer & Explanation

candelo6a

candelo6a

Beginner2022-06-14Added 24 answers

Your statement is equivalent to proving that, for
g ( x ) = 2 arctan 1 x + x 1
we also have
g ( x ) = π 2 arctan x 1
Note that
1 x + x 1 = x x 1
Set h ( x ) = x x 1 and prove that, for x>1, 0<h(x)<1. It follows that 0 < g ( x ) < π / 2
Now
tan g ( x ) = tan ( 2 arctan h ( x ) ) = 2 tan arctan ( h ( x ) ) 1 tan 2 ( arctan h ( x ) ) = 2 h ( x ) 1 ( h ( x ) ) 2 = 2 x x 1 1 x ( x 1 ) + 2 x ( x 1 ) = x x 1 x 1 ( x x 1 ) = 1 x 1
Therefore
tan ( π 2 g ( x ) ) = cot g ( x ) = x 1
and so
π 2 g ( x ) = arctan x 1

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