Discussion Forum

if abc= 1, then $\left(\frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}\right)=?$

Answer:

Explanation:

Given Exp. $=\left(\frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}\right)$ $=\frac{1}{1+a+b^{-1}}+\frac{b^{-1}}{b^{-1}+1+b^{-1}c^{-1}}+\frac{a}{a+ac+1}$ $=\frac{1}{1+a+b^{-1}}+\frac{b^{-1}}{1+b^{-1}+a}+\frac{a}{a+b^{-1}+1}$ $=\frac{1+a+b^{-1}}{1+a+b^{-1}}=1$ $\left[\because abc=1 \right] \Rightarrow \left(bc\right)^{-1}$ $=a\Rightarrow b^{-1}c^{-1}=a$ and $ac= b^{-1}$