Question

show that
$\frac{1}{1 + {a}^{y - x} + {a}^{z - x} } + \frac{1}{1 + {a}^{y - z} + {a}^{x - y} } + \frac{1}{1 + {a}^{x - z} + {a}^{y - z} }$
please uts urgent
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Answer 1

$\large\text{\underline{Proper Question}}$

Show that $\dfrac{1}{1+a^{y-x}+a^{z-x}}+\dfrac{1}{1+a^{z-y}+a^{x-y}}+\dfrac{1}{1+a^{x-z}+a^{y-z}}=1$.

$\large\text{\underline{Solution}}$

We are given the rational expression of the following;

$\dfrac{1}{1+a^{y-x}+a^{z-x}}+\dfrac{1}{1+a^{z-y}+a^{x-y}}+\dfrac{1}{1+a^{x-z}+a^{y-z}}$

Multiply $\dfrac{a^{x}}{a^{x}},\dfrac{a^{y}}{a^{y}},\dfrac{a^{z}}{a^{z}}$ into each fraction respectively;

$=\dfrac{a^{x}}{a^{x}+a^{y}+a^{z}}+\dfrac{a^{y}}{a^{y}+a^{z}+a^{x}}+\dfrac{a^{z}}{1+a^{x}+a^{y}}$

On simplifying,

$=\dfrac{a^{x}+a^{y}+a^{z}}{a^{x}+a^{y}+a^{z}}=1$