Question

If a + b + c = 0, Then Show That
$\frac{1}{ {x}^{b} + {x}^{ - c} + 1 } + \frac{1}{ {x}^{c} + {x}^{ - a} + 1 } \: + \: \frac{1}{ {x}^{a} + {x}^{ - b} + 1 } = 1$
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Answer 1

Given :--

• a + b + c = 0

To Show :-

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

Solution :-

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

Let us Solve by Taking LHS First 2 parts Individually first :-

→ 1/( x ^b + x ^ (-c) + 1 )

Multiply & Divide by x^(-b) we get,

→ [ 1/( x ^b + x ^ (-c) + 1 ) ] * [ x^(-b) / x^(-b) ]

Denominator Becomes :-

→ x^(-b) [ ( x ^b + x ^ (-c) + 1 ) ]

→ x^(-b + b) + x^(-b - c) + x^(-b) * 1 { using a^b * a^c = a^(b+c) }

→ x^0 + x^(-b - c) + x^(-b)

Now, Given that, a + b + c = 0, so, we get, a = ( - b - c) , Putting This value and x^0 = 1 , we get,

→ 1 + x^a + x^(-b)

Than our Entire Part becomes ,

→ x^(-b) / (x ^a + x^(-b) + 1 ) ---------------Equation (1) .

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Now, Similarly , Solving Second part :-

→ 1/( x ^ c + x ^(-a) + 1 )

Multiply & Divide by x^a This time we get,

→ [ 1/( x ^ c + x ^(-a) + 1 ) ] * (x^a / x^a)

→ [x^a / {x^a(x ^ c + x ^(-a) + 1 ) } ]

→ [x^a /{ x^(a + c) + 1 + x^a ) } ]

Now, a + b + c = 0 , so, a + c = (-b),,

→ [ x^a /{ x^(-b) + 1 + x^a ) } ]

→ x^a / (x ^a + x^(-b) + 1 ) ------------------ Equation (2)

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Putting both Value From Equation (1) & (2) Now, in Question we get,

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

Putting values :-

→ [ x^(-b) / (x ^a + x^(-b) + 1 ) ] + [x^a / (x ^a + x^(-b) + 1 )] + [ 1/( x ^a + x^(-b) + 1 ) ] = 1

Taking LCM of LHS, now we get, { since, Denominator are same in Each Now.}

→ [ ( x ^a + x^(-b) + 1 ) / ( x ^a + x^(-b) + 1 ) ] = 1

→ 1 = 1 (Hence, Proved).

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Answer 2

Answer:

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