Answer
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Hint: In this question use the given information to simplify the given equation and also remember to use the given equation xyz = 1 to find the value of y in terms of x and z, then put it in the LHS, using this information will help you to approach towards the solution of the question.
Complete step-by-step answer:
According to the given information we have xyz = 1 and we have to show that ${\left( {1 + x + {y^{ - 1}}} \right)^{ - 1}} + {\left( {1 + y + {z^{ - 1}}} \right)^{ - 1}} + {\left( {1 + z + {x^{ - 1}}} \right)^{ - 1}}$ is equal to 1
So taking equation ${\left( {1 + x + {y^{ - 1}}} \right)^{ - 1}} + {\left( {1 + y + {z^{ - 1}}} \right)^{ - 1}} + {\left( {1 + z + {x^{ - 1}}} \right)^{ - 1}}$ as L.H.S and 1 as R.H.S
Let’s simplify L.H.S
L.H.S = ${\left( {1 + x + {y^{ - 1}}} \right)^{ - 1}} + {\left( {1 + y + {z^{ - 1}}} \right)^{ - 1}} + {\left( {1 + z + {x^{ - 1}}} \right)^{ - 1}}$
$ \Rightarrow $L.H.S = ${\left( {1 + x + {y^{ - 1}}} \right)^{ - 1}} + {\left( {1 + y + {z^{ - 1}}} \right)^{ - 1}} + {\left( {1 + z + {x^{ - 1}}} \right)^{ - 1}}$
$ \Rightarrow $L.H.S = \[\dfrac{1}{{1 + x + {y^{ - 1}}}} + \dfrac{1}{{1 + y + {z^{ - 1}}}} + \dfrac{1}{{1 + z + {x^{ - 1}}}}\] (equation 1)
Since it is given that xyz = 1 which can be written as \[xz = {y^{ - 1}}\] and \[y = {\left( {xz} \right)^{ - 1}}\]
Substituting these values in the equation 1 we get
$ \Rightarrow $L.H.S = \[\dfrac{1}{{1 + x + xz}} + \dfrac{1}{{1 + {{\left( {xz} \right)}^{ - 1}} + {z^{ - 1}}}} + \dfrac{1}{{1 + z + {x^{ - 1}}}}\]
$ \Rightarrow $L.H.S = \[\dfrac{1}{{1 + x + xz}} + \dfrac{1}{{1 + \dfrac{1}{{xz}} + \dfrac{1}{z}}} + \dfrac{1}{{1 + z + \dfrac{1}{x}}}\]
$ \Rightarrow $L.H.S = \[\dfrac{1}{{1 + x + xz}} + \dfrac{1}{{\dfrac{{xz + 1 + x}}{{xz}}}} + \dfrac{1}{{\dfrac{{x + xz + 1}}{x}}}\]
$ \Rightarrow $L.H.S = \[\dfrac{1}{{1 + x + xz}} + \dfrac{{xz}}{{xz + 1 + x}} + \dfrac{x}{{x + xz + 1}}\]
Since the denominator of all the fractional equation is same therefore
L.H.S = \[\dfrac{{1 + x + xz}}{{1 + x + xz}}\]
So as we can see that numerator and denominator are same therefore
L.H.S = 1
Therefore L.H.S = R.H.S
Hence ${\left( {1 + x + {y^{ - 1}}} \right)^{ - 1}} + {\left( {1 + y + {z^{ - 1}}} \right)^{ - 1}} + {\left( {1 + z + {x^{ - 1}}} \right)^{ - 1}}$ is equal to 1.
Note: The above question was a very basic question which was based on the substitution method we used the given information xyz = 1 to find the value of ${y^{ - 1}}$ and y then we substituted the values in the equation and simplified it until the denominator became the same and then we used the addition operation and got the result 1.
Complete step-by-step answer:
According to the given information we have xyz = 1 and we have to show that ${\left( {1 + x + {y^{ - 1}}} \right)^{ - 1}} + {\left( {1 + y + {z^{ - 1}}} \right)^{ - 1}} + {\left( {1 + z + {x^{ - 1}}} \right)^{ - 1}}$ is equal to 1
So taking equation ${\left( {1 + x + {y^{ - 1}}} \right)^{ - 1}} + {\left( {1 + y + {z^{ - 1}}} \right)^{ - 1}} + {\left( {1 + z + {x^{ - 1}}} \right)^{ - 1}}$ as L.H.S and 1 as R.H.S
Let’s simplify L.H.S
L.H.S = ${\left( {1 + x + {y^{ - 1}}} \right)^{ - 1}} + {\left( {1 + y + {z^{ - 1}}} \right)^{ - 1}} + {\left( {1 + z + {x^{ - 1}}} \right)^{ - 1}}$
$ \Rightarrow $L.H.S = ${\left( {1 + x + {y^{ - 1}}} \right)^{ - 1}} + {\left( {1 + y + {z^{ - 1}}} \right)^{ - 1}} + {\left( {1 + z + {x^{ - 1}}} \right)^{ - 1}}$
$ \Rightarrow $L.H.S = \[\dfrac{1}{{1 + x + {y^{ - 1}}}} + \dfrac{1}{{1 + y + {z^{ - 1}}}} + \dfrac{1}{{1 + z + {x^{ - 1}}}}\] (equation 1)
Since it is given that xyz = 1 which can be written as \[xz = {y^{ - 1}}\] and \[y = {\left( {xz} \right)^{ - 1}}\]
Substituting these values in the equation 1 we get
$ \Rightarrow $L.H.S = \[\dfrac{1}{{1 + x + xz}} + \dfrac{1}{{1 + {{\left( {xz} \right)}^{ - 1}} + {z^{ - 1}}}} + \dfrac{1}{{1 + z + {x^{ - 1}}}}\]
$ \Rightarrow $L.H.S = \[\dfrac{1}{{1 + x + xz}} + \dfrac{1}{{1 + \dfrac{1}{{xz}} + \dfrac{1}{z}}} + \dfrac{1}{{1 + z + \dfrac{1}{x}}}\]
$ \Rightarrow $L.H.S = \[\dfrac{1}{{1 + x + xz}} + \dfrac{1}{{\dfrac{{xz + 1 + x}}{{xz}}}} + \dfrac{1}{{\dfrac{{x + xz + 1}}{x}}}\]
$ \Rightarrow $L.H.S = \[\dfrac{1}{{1 + x + xz}} + \dfrac{{xz}}{{xz + 1 + x}} + \dfrac{x}{{x + xz + 1}}\]
Since the denominator of all the fractional equation is same therefore
L.H.S = \[\dfrac{{1 + x + xz}}{{1 + x + xz}}\]
So as we can see that numerator and denominator are same therefore
L.H.S = 1
Therefore L.H.S = R.H.S
Hence ${\left( {1 + x + {y^{ - 1}}} \right)^{ - 1}} + {\left( {1 + y + {z^{ - 1}}} \right)^{ - 1}} + {\left( {1 + z + {x^{ - 1}}} \right)^{ - 1}}$ is equal to 1.
Note: The above question was a very basic question which was based on the substitution method we used the given information xyz = 1 to find the value of ${y^{ - 1}}$ and y then we substituted the values in the equation and simplified it until the denominator became the same and then we used the addition operation and got the result 1.
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