Question

$Prove space that space cot to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus sin space straight x end root plus square root of 1 minus sin space straight x end root over denominator square root of 1 plus sin space straight x end root minus square root of 1 minus sin space straight x end root end fraction close square brackets equals straight x over 2 comma space space straight x space element of space open parentheses 0 comma space straight pi over 4 close parentheses.$

Solution

straight L. straight H. straight S. space equals space cot to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus sin space straight x end root plus square root of 1 minus sin space straight x end root over denominator square root of 1 plus sin space straight x end root plus square root of 1 minus sin space straight x end root end fraction close square brackets space space space space space space space space space space space equals space cot to the power of negative 1 end exponent open square brackets fraction numerator square root of cos squared begin display style straight x over 2 end style plus sin squared begin display style straight x over 2 end style plus 2 space sin space begin display style straight x over 2 end style cos begin display style straight x over 2 end style end root plus square root of cos squared begin display style straight x over 2 end style plus sin squared begin display style straight x over 2 end style minus 2 space sin begin display style straight x over 2 end style cos begin display style straight x over 2 end style end root over denominator square root of cos squared straight x over 2 plus sin squared straight x over 2 plus 2 space sin space straight x over 2 cos straight x over 2 end root minus square root of cos squared straight x over 2 plus sin squared straight x over 2 minus 2 space sin straight x over 2 cos straight x over 2 end root end fraction close square brackets space space space space space space space space space space equals space cot to the power of negative 1 end exponent open square brackets fraction numerator square root of open parentheses cos begin display style straight x over 2 end style plus sin begin display style straight x over 2 end style close parentheses squared end root plus square root of open parentheses cos straight x over 2 minus sin straight x over 2 close parentheses squared end root over denominator square root of open parentheses cos straight x over 2 plus sin straight x over 2 close parentheses squared end root plus square root of open parentheses cos straight x over 2 minus sin straight x over 2 close parentheses squared end root end fraction close square brackets space space space space space space space space space space space space space space space space equals space open square brackets fraction numerator open parentheses cos straight x over 2 plus sin straight x over 2 close parentheses plus open parentheses cos straight x over 2 minus sin straight x over 2 close parentheses over denominator open parentheses cos straight x over 2 plus sin straight x over 2 close parentheses plus open parentheses cos straight x over 2 minus sin straight x over 2 close parentheses 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 space space space space space space space space space space space space open square brackets because space 0 space less than space straight x space less than straight pi over 2 space rightwards double arrow space 0 less than straight pi over 2 less than straight pi over 4 rightwards double arrow cos straight x over 2 greater than sin straight x over 2 close square brackets space space space space space space space space space space equals space cot to the power of negative 1 end exponent open parentheses fraction numerator 2 space cos space begin display style straight x over 2 end style over denominator 2 space sin space begin display style straight x over 2 end style end fraction close parentheses equals cot to the power of negative 1 end exponent open parentheses cot space straight x over 2 close parentheses equals straight x over 2 space space space space space space space space space space equals space straight R. straight H. straight S. space

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