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if \: x =( √(3) )/(2) then \: find \: \ \\ the \: \\ value \: of \\ \\ \\ \ \: ( √(1 + x) )/(1 + √(1 + x) ) + ( √(1 - x) )/(1 - √(1 - x) )

If x=underoot 3/2,then find the value of underoot 1+x by 1+ underoot 1+x +underoot 1-x by 1- underoot 1-x.​

User Debjani
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1 Answer

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\bf \underline{Given-} \\


\rm{x = ( √(3) )/(2) } \\


\bf \underline{To \: find-} \\


\mathsf{ (1 + x)/(1 + √(1 + x) ) + (1 - x)/(1 - √(1 - x) ) = ?} \\


\bf \underline{Solution-} \\


\rm{ √(1 + x) } \\


= \sqrt{1 + ( √(3) )/(2) } \:\:\:\: \: \: \: \sf{ [∵\: \:x = √3/2\:(Given)]}\\


= \sqrt{ ( 2 + √(3) )/(2) } \\


= \sqrt{ (4 + 2 √(3) )/(4) } \\


= \sqrt{ (3 + 1 + 2 * √(3) * 1)/(4) } \\


= \sqrt{ \frac{( √(3) + 1 {)}^(2) }{4} } \\


= ( √(3) + 1 )/(2) \\


\textsf{Similarly,} \\


\rm{ √(1 - x) } \\


\rm{ = ( √(3) - 1)/(2) } \\


\rm{\therefore \: (1 + x)/(1 + √(1 + x) ) + (1 - x)/(1 - √(1 - x) ) } \\


\rm{= (1 + ( √(3) )/(2) )/(1 + ( √(3) + 1 )/(2) ) + ( 1 - ( √(3) )/(2) )/(1 - \bigg( ( √(3) - 1 )/(2) \bigg) ) } \\


\rm{= (2 + √(3) )/(2 + √(3) + 1 ) + (2 - √(3) )/(2 - ( √(3) - 1)) } \\


\rm{= (2 + √(3) )/(3 + √(3) ) + (2 - √(3) )/(3 - √(3) ) } \\


\rm{= (2 + √(3) )/(3 + √(3) ) * (3 - √(3) )/(3 - √(3) ) + (2 - √(3) )/(3 - √(3) ) * (3 + √(3) )/(3 + √(3) ) } \\


\rm{= \frac{(2 + √(3) )(3 - √(3) ) + (2 - √(3))(3 + √(3) )}{(3 {)}^(2) - ( √(3) {)}^(2) } } \\


\rm{= (6 - 2 √(3) + 3 √(3) - 3 + 6 + 2 √(3) - 3 √(3) - 3)/(9 - 3) } \\


\rm{= (12 - 6)/(6) } \\


\rm{= (6)/(6) } \\


\rm{= 1} \\


\bf \underline{Answer-} \\


\mathsf{\underline{Hence, the \: value \: of \: (1 + x)/(1 + √(1 + x) ) + (1 - x)/(1 - √(1 - x) ) \: is \: 1}} \\

User Timothy Fisher
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