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Solve for X : Tan − 1 ( 2 − X 2 + X ) = 1 2 Tan − 1 X 2 , X > 0 . - Mathematics

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Sum

Solve for x : `tan^-1 ((2-"x")/(2+"x")) = (1)/(2)tan^-1  ("x")/(2), "x">0.`

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Solution

Given that

⇒ `tan^-1 ((2-"x")/(2+"x")) = (1)/(2) tan^-1  ("x")/(2)`

⇒ `2tan^-1 ((2-"x")/(2+"x")) = tan^-1  ("x")/(2)`

⇒ `tan^-1  (2((2-"x")/(2+"x")))/(1 - ((2-"x")/(2+"x"))^2)  = tan^-1  ("x")/(2)` 

⇒ `tan^-1  (4 - x^2)/(4x) = tan^-1  ("x")/(2)`

⇒ `(4 -"x"^2)/(4"x") = ("x")/(2)`

`"x" = 2/sqrt3   ...[∵ "x" >0]`.

Concept: Properties of Inverse Trigonometric Functions
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