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Question
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]`.
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