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Yesenia Sherman 2022-06-24 Answered
How to simplify [ 2 tan 1 x i ] ?
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Answers (1)

enfujahl
Answered 2022-06-25 Author has 20 answers
[ 2 tan 1 x i ] = [ 2 tan 1 x e i π / 4 ] = ( 2 tan 1 x e i π / 4 ) + ( 2 tan 1 x e i π / 4 ) 2
= 1 2 ( tan 1 x e i π / 4 + tan 1 x e i π / 4 ) = 1 2 tan 1 x e i π / 4 + x e i π / 4 1 x e i π / 4 x e i π / 4
= 1 2 tan 1 2 x cos π 4 1 x 2 = 1 2 tan 1 2 x 1 x 2
where line 2 3 is achieved by using that tan ( x + y ) = tan x + tan y 1 tan x tan y
The differs from the expression you give by a constant, after further noting that tan 1 x + tan 1 ( 1 x ) = π 2 for x>0
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