Question

$Prove space that space fraction numerator 9 straight pi over denominator 8 end fraction minus 9 over 4 space sin to the power of negative 1 end exponent 1 third equals 9 over 4 sin to the power of negative 1 end exponent fraction numerator 2 square root of 2 over denominator 3 end fraction.$

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Solution

straight L. straight H. straight S. space equals space fraction numerator 9 straight pi over denominator 8 end fraction minus 9 over 4 sin to the power of negative 1 end exponent open parentheses 1 third close parentheses equals 9 over 4 open curly brackets straight pi over 2 minus sin to the power of negative 1 end exponent 1 third close curly brackets space space space space space space space space space space equals space 9 over 4 space open curly brackets cos to the power of negative 1 end exponent open parentheses 1 third close parentheses close curly 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 open square brackets because space sin to the power of negative 1 end exponent space straight x plus space cos to the power of negative 1 end exponent space straight x equals straight x over 2 close square brackets space space space space space space space space space space equals 9 over 4 space sin to the power of negative 1 end exponent square root of 1 minus open parentheses 1 third close parentheses squared end root space space space space space space space space space space space space left square bracket because space cos to the power of negative 1 end exponent space straight x space equals space sin to the power of negative 1 end exponent space square root of 1 minus straight x squared end root space for space 0 less or equal than straight x space less or equal than space 1 space right square bracket space space space space space space space space space space equals space 9 over 4 space sin to the power of negative 1 end exponent space open parentheses square root of 8 over 9 end root close parentheses equals 9 over 4 space sin to the power of negative 1 end exponent space open parentheses fraction numerator 2 square root of 2 over denominator 3 end fraction close parentheses equals space straight R. straight H. straight S.

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Inverse Trigonometric Functions

$Prove space that space fraction numerator 9 straight pi over denominator 8 end fraction minus 9 over 4 space sin to the power of negative 1 end exponent 1 third equals 9 over 4 sin to the power of negative 1 end exponent fraction numerator 2 square root of 2 over denominator 3 end fraction.$

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