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प्रश्न
Prove that:
`1/(1 + x^(b - a) + x^(c - a)) + 1/(1 + x^(a - b) + x^(c - b)) + 1/(1 + x^(b - c) + x^(a - c)) = 1`
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उत्तर
Consider the left hand side:
`1/(1 + x^(b - a) + x^(c - a)) + 1/(1 + x^(a - b) + x^(c - b)) + 1/(1 + x^(b - c) + x^(a - c))`
= `1/(1 + x^b xx x^-a + x^c xx x^-a) + 1/(1 + x^a xx x^-b + x^c xx x^-b) + 1/(1 + x^b xx x^-c + x^a xx x^-c)` ...[∵ am + n = am × an]
= `1/(1 + x^b/x^a + x^c/x^a) + 1/(1 + x^a/x^b + x^c/x^b) + 1/(1 + x^b/x^c + x^a/x^c)`
= `1/((x^a + x^b + x^c)/x^a) + 1/((x^b + x^a + x^c)/x^b) + 1/((x^c + x^b + x^a)/x^c)`
= `x^a/(x^a + x^b + x^c) + x^b/(x^a + x^b + x^c) + x^c/(x^a + x^b + x^c)`
= `(x^a + x^b + x^c)/(x^a + x^b + x^c)`
= 1
Therefore left hand side is equal to the right hand side.
Hence proved.
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