If abc = 1, then $${\frac{1}{{1 + a + {b^{ - 1}}}} + }$$   $${\frac{1}{{1 + b + {c^{ - 1}}}} + }$$   $${\frac{1}{{1 + c + {a^{ - 1}}}}}$$   = ?

Given expression,
$${\frac{1}{{1 + a + {b^{ - 1}}}} + }$$   $${\frac{1}{{1 + b + {c^{ - 1}}}} + }$$   $${\frac{1}{{1 + c + {a^{ - 1}}}}}$$
  $$ = \frac{1}{{1 + a + {b^{ - 1}}}} + $$   $$\frac{b^{ - 1}}{{{b^{ - 1}} + 1 + {b^{ - 1}}{c^{ - 1}}}} + $$    $$\frac{1}{{a + ac + 1}}$$
  $$ = \frac{1}{{1 + a + {b^{ - 1}}}} + $$   $$\frac{{{b^{ - 1}}}}{{1 + {b^{ - 1}} + a}} + $$   $$\frac{a}{{a + {b^{ - 1}} + 1}}$$
$$\eqalign{ & = \frac{{1 + a + {b^{ - 1}}}}{{1 + a + {b^{ - 1}}}} \cr & = 1 \cr} $$
$$\left[ {\because abc = 1 \Rightarrow {{\left( {bc} \right)}^{ - 1}} = a \Rightarrow {b^{ - 1}}{c^{ - 1}} = a,{\text{and }}ac = {b^{ - 1}}} \right]$$