Question

Prove that
If $tan to the power of negative 1 end exponent open parentheses fraction numerator straight x minus 2 over denominator straight x minus 4 end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 2 over denominator straight x plus 4 end fraction close parentheses equals straight pi over 4 comma$ find the value of x.

Solution

$Given space that space tan to the power of negative 1 end exponent open parentheses fraction numerator straight x minus 2 over denominator straight x minus 4 end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 2 over denominator straight x plus 4 end fraction close parentheses equals straight pi over 4 We space need space to space find space the space value space of space straight x. tan to the power of negative 1 end exponent open parentheses fraction numerator straight x minus 2 over denominator straight x minus 4 end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 2 over denominator straight x plus 4 end fraction close parentheses equals straight pi over 4 rightwards double arrow space space tan to the power of negative 1 end exponent open parentheses fraction numerator begin display style fraction numerator straight x minus 2 over denominator straight x minus 4 end fraction plus fraction numerator straight x plus 2 over denominator straight x plus 4 end fraction end style over denominator 1 minus open parentheses begin display style fraction numerator straight x minus 2 over denominator straight x minus 4 end fraction end style close parentheses open parentheses begin display style fraction numerator straight x plus 2 over denominator straight x plus 4 end fraction end style close parentheses end fraction close parentheses space equals space tan straight pi over 4 rightwards double arrow space fraction numerator begin display style fraction numerator straight x minus 2 over denominator straight x minus 4 end fraction end style plus begin display style fraction numerator straight x plus 2 over denominator straight x plus 4 end fraction end style over denominator 1 minus open parentheses begin display style fraction numerator straight x minus 2 over denominator straight x minus 4 end fraction end style close parentheses open parentheses begin display style fraction numerator straight x plus 2 over denominator straight x plus 4 end fraction end style close parentheses end fraction equals tan straight pi over 4 rightwards double arrow space space fraction numerator left parenthesis straight x minus 2 right parenthesis left parenthesis straight x plus 4 right parenthesis plus left parenthesis straight x plus 2 right parenthesis left parenthesis straight x minus 4 right parenthesis over denominator left parenthesis straight x minus 4 right parenthesis left parenthesis straight x plus 4 right parenthesis minus left parenthesis straight x minus 2 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction equals 1 rightwards double arrow space fraction numerator left parenthesis straight x squared plus 2 straight x minus 8 right parenthesis plus left parenthesis straight x squared minus 2 straight x minus 8 right parenthesis over denominator left parenthesis straight x squared minus 16 right parenthesis minus left parenthesis straight x squared minus 4 right parenthesis end fraction equals 1 rightwards double arrow fraction numerator 2 straight x squared minus 16 over denominator negative 12 end fraction equals 1 rightwards double arrow 2 straight x squared minus 16 equals negative 12 rightwards double arrow 2 straight x squared space equals 4 rightwards double arrow straight x squared space equals 2 rightwards double arrow space straight x equals plus-or-minus square root of 2$

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