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Inverse Trigonometric Functions

Question
CBSEENMA12032770
Wired Faculty App

Show that sin to the power of negative 1 end exponent 12 over 13 plus cos to the power of negative 1 end exponent 4 over 5 plus tan to the power of negative 1 end exponent 63 over 16 equals straight pi.

Solution
Let space sin to the power of negative 1 end exponent 12 over 13 equals straight x comma space cos to the power of negative 1 end exponent 4 over 5 equals straight y comma space space space tan to the power of negative 1 end exponent 63 over 16 equals straight z therefore space space space sin space straight x equals 12 over 13 comma space cos space straight y equals 4 over 5 comma space space tan space straight z equals 63 over 16 therefore space space space cos space straight x equals square root of 1 minus sin squared space straight x end root equals square root of 1 minus 144 over 169 end root equals square root of 25 over 169 end root equals 5 over 13 therefore space space space tan space straight x equals fraction numerator sin space straight x over denominator cos space straight x end fraction equals 12 over 13 cross times 13 over 5 equals 12 over 5 Again space sin space straight y space equals space square root of 1 minus cos squared space straight y end root equals square root of 1 minus 16 over 25 end root equals square root of 9 over 25 end root equals 3 over 5 therefore space space space tan space straight y space equals space fraction numerator sin space straight y over denominator space cos space straight y end fraction equals 3 over 5 cross times 5 over 4 equals 3 over 4 therefore space space cos space straight x space equals space 5 over 13 comma space sin space straight y equals 3 over 5 comma space tan space straight x space equals 12 over 5 space and space tan space straight y space equals space 3 over 4 Now space space space tan space left parenthesis straight x plus straight y right parenthesis equals fraction numerator tan space straight x plus tan space straight y over denominator 1 asterisk times tan space straight x space tan space straight y end fraction equals fraction numerator begin display style 12 over 5 end style plus begin display style 3 over 4 end style over denominator 1 minus begin display style 12 over 5 end style cross times begin display style 3 over 4 end style end fraction equals fraction numerator 48 plus 15 over denominator 20 minus 36 end fraction equals negative 63 over 16 therefore space space space space space space space space space space space space space tan space space left parenthesis straight x plus straight y right parenthesis space space space equals space minus space tan space straight z where space space space space space space space space tan space straight z equals 63 over 16 space space space straight i. straight e. space space space straight z equals space tan to the power of negative 1 end exponent open parentheses 63 over 16 close parentheses
i.e.  tan (x + y) = tan (- z) or tan (x + y) = tan open parentheses straight pi minus straight z close parentheses
therefore   x + y = - z or x + y = straight pi minus straight z
Since x, y and x are positive , straight x plus straight y not equal to negative straight z
therefore   x + y + z = straight pi
or space space space space space space sin to the power of negative 1 end exponent 12 over 13 plus cos to the power of negative 1 end exponent 4 over 5 plus tan to the power of negative 1 end exponent 63 over 16 equals straight pi

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