Question

If abc = 1, show that `1/(1+a+b^-1)+1/(1+b+c^-1)+1/(1+c+a^-1)=1`

Answer 1

Consider the left hand side:

`1/(1+a+b^-1)+1/(1+b+c^-1)+1/(1+c+a^-1)`

`=1/(1+a+1/b)+1/(1+b+1/c)+1/(1+c+1/a)`

`=1/((b+ab+1)/b)+1/((c+bc+1)/c)+1/((a+ac+1)/a)`

`=b/(b+ab+1)+c/(c+bc+1)+a/(a+ac+1)`        ...........(1)

We know that abc = 1

`therefore c = 1/(ab)`

By substituting the value of c in equation (1), we get

`=b/(b+ab+1)+(1/(ab))/(1/(ab)+b(1/(ab))+1)+a/(a+a(1/(ab))+1)`

`=b/(b+ab+1)+(1/(ab))/(1/(ab)+b/(ab)+(ab)/(ab))+a/((ab)/b+1/b+b/b)`

`=b/(b+ab+1)+(1/(ab))/((1+b+ab)/(ab))+a/((ab+1+b)/(b))`

`=b/(b+ab+1)+(1/(ab)xxab)/(1+b+ab)+(axxb)/(ab+1+b)`

`=b/(b+ab+1)+1/(b+ab+1)+(ab)/(b+ab+1)`

`=(b+ab+1)/(b+ab+1)`

= 1

Therefore, LHS = RHS

Hence, proved