Question

$tan to the power of negative 1 end exponent fraction numerator x minus 1 over denominator x minus 2 end fraction plus tan to the power of negative 1 end exponent fraction numerator x plus 1 over denominator x plus 2 end fraction equals straight pi over 4.$

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Solution

tan to the power of negative 1 end exponent fraction numerator straight x minus 1 over denominator straight x minus 2 end fraction plus tan to the power of negative 1 end exponent fraction numerator straight x plus 1 over denominator straight x plus 2 end fraction equals straight pi over 4 rightwards double arrow space space tan to the power of negative 1 end exponent open square brackets fraction numerator begin display style fraction numerator straight x minus 1 over denominator straight x plus 2 end fraction end style plus begin display style fraction numerator straight x plus 1 over denominator straight x plus 2 end fraction end style over denominator 1 minus begin display style fraction numerator straight x minus 1 over denominator straight x plus 2 end fraction end style cross times begin display style fraction numerator straight x plus 1 over denominator straight x plus 2 end fraction end style end fraction close square brackets equals straight pi over 4 rightwards double arrow space space space fraction numerator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis plus left parenthesis straight x minus 2 right parenthesis left parenthesis straight x plus 1 right parenthesis over denominator left parenthesis straight x minus 2 right parenthesis left parenthesis straight x plus 2 right parenthesis minus left parenthesis straight x minus 1 right parenthesis left parenthesis straight x plus 1 right parenthesis end fraction equals tan straight pi over 4 space space space space space space rightwards double arrow space space space fraction numerator straight x squared plus straight x minus 2 plus straight x squared minus straight x minus 2 over denominator straight x squared minus 4 minus straight x squared plus 1 end fraction rightwards double arrow space space fraction numerator 2 space straight x squared minus 4 over denominator negative 3 end fraction equals 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space 2 space straight x squared minus 4 equals negative 3 space space space rightwards double arrow space 2 space straight x squared equals 1 rightwards double arrow space space straight x squared equals 1 half space space space space space rightwards double arrow space straight x equals 1 plus-or-minus fraction numerator 1 over denominator square root of 2 end fraction

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Inverse Trigonometric Functions

$tan to the power of negative 1 end exponent fraction numerator x minus 1 over denominator x minus 2 end fraction plus tan to the power of negative 1 end exponent fraction numerator x plus 1 over denominator x plus 2 end fraction equals straight pi over 4.$

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