Question

if a+b+c=0 then show that

Answer 1

given,

a + b + c = 0

1/( x^b + x^-c +1) + 1/( x^c + x^-a +1) + 1/( x^a + x^-b +1) = 1

step 1 :-

1/( x^b + x^-c +1) = 1/( x^b + 1/x^c + 1)

=x^c/{ x^( b+ c) + 1 + x^c }

= x^c/{ x^-a + 1 + x^c }

{ a + b + c =0 so, b + c = -a

=x^c/{ 1/x^a + 1 + x^c }

=x^( c + a)/{ 1 + x^a + x^( c + a) }

= x^-b/( 1 + x^a + x^-b ) -------(1)


step 2 :-
1/( x^c + x^-a + 1) = 1/( x^c + 1/x^a + 1 )

= x^a/{ x^( c + a) + 1 + x^a }

=x^a/( x^-b + 1 + x^a ) ------(2)


step 3 :- put equation (1) and (2) in
1/( x^b + x^-c +1) +1/( x^c + x^-a +1) +1/( x^a + x^-b +1)

e.g

x^-b/( 1+ x^a + x^-b) + x^a /( x^-b + 1 +x^a )
+ 1/( x^a + x^-b + 1)

=( x^a + x^-b+1)/( x^a +x^-b +1)

=1

hence proved ///