Question

prove that::

$\begin{gathered} \tt \: \frac{1}{1 + {x}^{b - a} + {x}^{c - a} } + \frac{1}{1 + {x}^{a - b} + {x}^{c - b} } + \frac{1}{1 + {x}^{b - c} {x}^{a - c} }=\end{gathered}$​

Answer 1

$\huge\star\underline{\mathtt\orange{Q} \mathfrak\blue{u }\mathfrak\blue{E} \mathbb\purple{ s}\mathtt\orange{T} \mathbb\pink{iOn}}\star$

$\begin{gathered} \tt \: \frac{1}{1 + {x}^{b - a} + {x}^{c - a} } + \frac{1}{1 + {x}^{a - b} + {x}^{c - b} } + \frac{1}{1 + {x}^{b - c} {x}^{a - c} }=\end{gathered}$

${\huge{\fcolorbox{aqua}{navy}{\fcolorbox{yellow}{blue}{\bf{\color{yellow}{⫷AnSwEr⫸}}}}}}$

$\begin{gathered}\rightsquigarrow \tt \: \frac{1}{1 + {x}^{b - a} + {x}^{c - a} } + \frac{1}{1 + {x}^{a - b} + {x}^{c - b} } + \frac{1}{1 + {x}^{b - c} {x}^{a - c} } \\ \\ \\ \\ : \rightsquigarrow \: \tt \: \frac{ {x}^{a} }{ {x}^{a} \bigg(1 + {x}^{b - a} + {x}^{c - a} \bigg) } + \frac{ {x}^{b} }{ {x}^{b} \bigg(1 + {x}^{a - b} + {x}^{c - b} \bigg)} + \frac{ {x}^{c} }{ {x}^{c} \bigg( 1 + {x}^{b - c} + {x}^{a - c} \bigg) } \\ \\ \\ \\ : \rightsquigarrow\tt \frac{ {x}^{a} }{ {x}^{a} + {x}^{b} + {x}^{c} } + \frac{ {x}^{b} }{ {x}^{b} + {x}^{a} + {x}^{c} } + \frac{ {x}^{c} }{ {x}^{c} + {x}^{b} + {x}^{a} } \\ \\ \\ \\ : \rightsquigarrow \tt \bigg(\frac{ \cancel{{x}^{a} + {x}^{b} + {x}^{c}} }{ \cancel{ {x}^{a} + {x}^{b} + {x}^{c} } } \bigg) \\ \\ \\ \\ \bf \: \large : \rightsquigarrow \large \: {\tt 1 }\end{gathered}$

Answer 2

Step-by-step explanation:

$\begin{gathered} \tt \: \frac{1}{1 + {x}^{b - a} + {x}^{c - a} } + \frac{1}{1 + {x}^{a - b} + {x}^{c - b} } + \frac{1}{1 + {x}^{b - c} {x}^{a - c} } \\ \\ \\ \\ : \implies \: \tt \: \frac{ {x}^{a} }{ {x}^{a} \bigg(1 + {x}^{b - a} + {x}^{c - a} \bigg) } + \frac{ {x}^{b} }{ {x}^{b} \bigg(1 + {x}^{a - b} + {x}^{c - b} \bigg)} + \frac{ {x}^{c} }{ {x}^{c} \bigg( 1 + {x}^{b - c} + {x}^{a - c} \bigg) } \\ \\ \\ \\ : \implies \tt \frac{ {x}^{a} }{ {x}^{a} + {x}^{b} + {x}^{c} } + \frac{ {x}^{b} }{ {x}^{b} + {x}^{a} + {x}^{c} } + \frac{ {x}^{c} }{ {x}^{c} + {x}^{b} + {x}^{a} } \\ \\ \\ \\ : \implies \tt \bigg(\frac{ \cancel{{x}^{a} + {x}^{b} + {x}^{c}} }{ \cancel{ {x}^{a} + {x}^{b} + {x}^{c} } } \bigg) \\ \\ \\ \\ \bf \: \large : \implies \large \: {\tt 1 \: \: \: \: }\end{gathered}$