If a+b+c=0,prove that:1/(a+x^a+x^-b)+1/(a+x^b+x^-c)+1/(1+x^c+x^-a)
If a+b+c=0,
prove that:
1/(a+x^a+x^-b)+1/(a+x^b+x^-c)+1/(1+x^c+x^-a)
Aayush , 7 Years ago
Grade 9
9 grade maths> If a+b+c=0, prove that: 1/(a+x^a+x^-b)+1/...
1 Answers
Arun
Dear Ayush
Reverse Proof-
1/(x^b + x^(-c) + 1) + 1/(x^c+ x^(-a)+1) + 1/ (x^a + x^(-b) + 1) = 1
1/(x^b + x^(-c) + 1) + 1/(x^c+ x^(-a)+1) + 1/ (x^a + x^(-b) + 1) - 1 = 0
[x^(a+b+c) - 1]^2/[(x^(a + b) + x^b +1) (x^(a + c) + x^a + 1) (x^(b + c) + x^c + 1)] = 0
x^(a + b + c) = 1
a + b + c = 0
1/(x^b + x^(-c) + 1) + 1/(x^c+ x^(-a)+1) + 1/ (x^a + x^(-b) + 1) - 1 = 0
[x^(a+b+c) - 1]^2/[(x^(a + b) + x^b +1) (x^(a + c) + x^a + 1) (x^(b + c) + x^c + 1)] = 0
x^(a + b + c) = 1
a + b + c = 0
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