Question

Let $tan to the power of negative 1 end exponent space straight y space equals tan to the power of negative 1 end exponent space straight x space plus space tan to the power of negative 1 end exponent space open parentheses fraction numerator 2 straight x over denominator 1 minus straight x squared end fraction close parentheses comma$ where $vertical line straight x vertical line space less than space fraction numerator 1 over denominator square root of 3 space end fraction space$ .Then, a value of y is

$fraction numerator 3 straight x squared minus straight x squared over denominator 1 minus 3 straight x squared end fraction$
$fraction numerator 3 straight x space plus space straight x cubed over denominator 1 minus 3 straight x squared end fraction$
$fraction numerator 3 straight x minus straight x cubed over denominator 1 plus 3 straight x squared end fraction$
$fraction numerator 3 straight x space plus space straight x cubed over denominator 1 plus 3 straight x squared end fraction$

Solution

fraction numerator 3 straight x squared minus straight x squared over denominator 1 minus 3 straight x squared end fraction

vertical line straight x vertical line less than space fraction numerator 1 over denominator square root of 3 end fraction space rightwards double arrow negative fraction numerator 1 over denominator square root of 3 end fraction less than straight x space less than thin space fraction numerator 1 over denominator square root of 3 end fraction Let space straight x space equals space tan space straight theta rightwards double arrow fraction numerator negative straight pi over denominator 6 end fraction space less than space straight theta space less than thin space straight pi over 6 tan to the power of negative 1 end exponent space straight y space equals space straight theta space plus space tan to the power of negative 1 end exponent space left parenthesis tan space 2 straight theta right parenthesis space equals space straight theta space plus space 2 straight theta space equals space 3 straight theta rightwards double arrow space straight y space equals space tan space 3 straight theta straight y equals space fraction numerator 3 space tan space straight theta minus tan cubed space straight theta over denominator 1 minus 3 space tan squared straight theta end fraction straight y space equals space fraction numerator 3 straight x minus straight x cubed over denominator 1 minus 3 straight x squared end fraction

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