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Mathematics

Prove that :

a+b+ca−1b−1+b−1c−1+c−1a−1=abc\dfrac{a + b + c}{a^{-1} b^{-1} + b^{-1}c^{-1} + c^{-1}a^{-1}} = abc

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Answer

To prove:

a+b+ca−1b−1+b−1c−1+c−1a−1=abc\dfrac{a + b + c}{a^{-1} b^{-1} + b^{-1}c^{-1} + c^{-1}a^{-1}} = abc

Solving L.H.S. of the above equation, we get :

⇒a+b+ca−1b−1+b−1c−1+c−1a−1=a+b+c1ab+1bc+1ca=a+b+cc+a+babc=abc(a+b+c)a+b+c=abc.\Rightarrow \dfrac{a + b + c}{a^{-1} b^{-1} + b^{-1}c^{-1} + c^{-1}a^{-1}} = \dfrac{a + b + c}{\dfrac{1}{ab} + \dfrac{1}{bc} + \dfrac{1}{ca}} \\[1em] = \dfrac{a + b + c}{\dfrac{c + a + b}{abc}} \\[1em] = \dfrac{abc(a + b + c)}{a + b + c} \\[1em] = abc.

Since, L.H.S. = R.H.S. = abc.

Hence, proved that a+b+ca−1b−1+b−1c−1+c−1a−1=abc\dfrac{a + b + c}{a^{-1} b^{-1} + b^{-1}c^{-1} + c^{-1}a^{-1}} = abc.

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