Question

If xyz=1, prove that
$\frac{1}{1 + x + {y}^{ - 1}} + \frac{1}{1 + y + {z}^{ - 1} } + \frac{1}{1 + z + {x}^{ - 1} } = 1$
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Answer 1

Answer is 1

Step-by-step explanation:

Given that xyz=1

⟹y−1=xz and ⟹y=(xz)−1 .

The key idea is to write the y in terms of x,z .

Now,

1/(1+x+y-¹)+1/(1+y+z−¹)+1/(1+z+x−¹)

=1/(1+x+xz)+1/(1+(xz)−¹+z−¹)+1/(1+z+x−¹)

=1/(1+x+xz)+xz/(xz+1+x)+x/(x+xz+1)

=1/(1+x+xz)+xz/(1+x+xz)+x/(1+x+xz)

=1+x+xz/(1+x+xz)

=1