Question

if abc=1 then show that
$\frac{1}{1 + a + {b}^{ - 1} } + \frac{1}{1 + b + {c}^{ - 1} } + \frac{1}{1 +c + {a}^{ - 1} } = 1$
​

Answer 1

Lets solve it with a short method....

Concept

If there are cyclic factors then, it will have roots

equal to a = 1, b = 1 and c = 1.

Cyclic factors

If we get the same question after replacing a as b, b as c and c as a, then a, b and c are cyclic factors.

Applying

Replacing all 'a' in question with 'b', all 'b' with 'c'

and all 'c' with 'a'.

We see that we get original question again.

It means a, b and c are cyclic roots and then a = 1,

b = 1 and c = 1.

Solving

Putting values of a, b and c in question i.e. 1.

We get,

1/3 + 1/3 + 1/3 = 1

LHS = RHS

Hence, proved.

Thank you.