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Inverse Trigonometric Functions

Question

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Show that $tan to the power of negative 1 end exponent 1 half plus tan to the power of negative 1 end exponent 2 over 11 equals tan to the power of negative 1 end exponent 3 over 4.$

Solution

tan to the power of negative 1 end exponent 1 half plus tan to the power of negative 1 end exponent 2 over 11 equals tan to the power of negative 1 end exponent open square brackets fraction numerator begin display style 1 half plus 2 over 11 end style over denominator 1 minus begin display style 1 half end style cross times begin display style 2 over 11 end style end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator begin display style fraction numerator 11 plus 4 over denominator 22 end fraction end style over denominator begin display style fraction numerator 22 minus 2 over denominator 22 end fraction end style end fraction close square brackets 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 equals space tan to the power of negative 1 end exponent open parentheses 15 over 20 close parentheses equals space tan to the power of negative 1 end exponent open parentheses 3 over 4 close parentheses space rightwards double arrow space straight R. straight H. straight S. therefore space space space space straight L. straight H. straight S. space equals space straight R. straight H. straight S

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