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1 / { 1 + x^( a - b )} + 1 / { 1 + x^(b - a )} = 1
LHS = 1 / { 1 + x^( a - b )} + 1 / { 1 + x^-( a - b )}
=> 1 / { 1 + x^( a - b )} + 1 / { 1 + 1 / x^( a- b )}
=>[ 1 / { 1 + x^( a - b )}] + [ 1 / { x^ (a - b ) + 1 } / x^( a - b ) ]
=> [ 1 / { 1 + x^( a - b )}] + [ x^ ( a - b ) / x^ ( a - b ) +1 ]
=> [ 1 + x^ ( a - b ) ] / [ 1 + x^ ( a - b )]
=> 1
=> R. H. S.
Hence, PROVED.
LHS = 1 / { 1 + x^( a - b )} + 1 / { 1 + x^-( a - b )}
=> 1 / { 1 + x^( a - b )} + 1 / { 1 + 1 / x^( a- b )}
=>[ 1 / { 1 + x^( a - b )}] + [ 1 / { x^ (a - b ) + 1 } / x^( a - b ) ]
=> [ 1 / { 1 + x^( a - b )}] + [ x^ ( a - b ) / x^ ( a - b ) +1 ]
=> [ 1 + x^ ( a - b ) ] / [ 1 + x^ ( a - b )]
=> 1
=> R. H. S.
Hence, PROVED.
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