Inverse Trigonometric Functions

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Question

Prove that : $tan to the power of negative 1 end exponent open parentheses 1 half close parentheses plus tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses equals straight pi over 4$

Solution

straight L. straight H. straight S. equals space tan to the power of negative 1 end exponent open parentheses 1 half close parentheses plus tan to the power of negative 1 end exponent open parentheses 1 fifth close parentheses plus tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses space space space space space space space space space space equals open square brackets tan to the power of negative 1 end exponent open parentheses 1 half close parentheses plus tan to the power of negative 1 end exponent open parentheses 15 over blank close parentheses close square brackets plus tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses space space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets fraction numerator begin display style 1 half end style plus begin display style 1 fifth end style over denominator 1 minus open parentheses begin display style 1 half end style close parentheses open parentheses begin display style 1 fifth end style close parentheses end fraction close square brackets plus tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses 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 space space space space space space space open square brackets because space tan to the power of negative 1 end exponent straight x plus tan to the power of negative 1 end exponent straight y equals tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus straight y over denominator 1 minus xy end fraction close parentheses close square brackets space space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets fraction numerator 5 plus 2 over denominator 10 minus 1 end fraction close square brackets plus tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses equals tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses space space space space space space space space space space equals space tan to the power of negative 1 end exponent space open square brackets fraction numerator begin display style 7 over 9 end style plus begin display style 1 over 8 end style over denominator 1 minus open parentheses begin display style 7 over 9 end style close parentheses open parentheses begin display style 1 over 8 end style close parentheses end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator 56 plus 9 over denominator 72 minus 7 end fraction close square brackets equals tan to the power of negative 1 end exponent open parentheses 65 over 65 close parentheses space space space space space space space space space space equals space tan to the power of negative 1 end exponent left parenthesis 1 right parenthesis equals straight pi over 4 equals straight R. straight H. straight S.

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Inverse Trigonometric Functions

Prove that : $tan to the power of negative 1 end exponent open parentheses 1 half close parentheses plus tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses equals straight pi over 4$

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