$\left| {\begin{array}{{}{c}}{{{\left( {{a^x} + {x^{ - x}}} \right)}^2} - {{\left( {{a^x} - {a^{ - x}}} \right)}^2}}&{{{\left( {{a^x} - {a^{ - x}}} \right)}^2}}&1\\{{{\left( {{b^y} + {b^{ - y}}} \right)}^2} - {{\left( {{b^y} - {b^{ - y}}} \right)}^2}}&{{{\left( {{b^y} - {b^{ - y}}} \right)}^2}}&1\\{{{\left( {{c^z} + {c^{ - z}}} \right)}^2} - {{\left( {{c^z} - {c^{ - z}}} \right)}^2}}&{{{\left( {{c^z} - {c^{ - z}}} \right)}^2}}&1\end{array}} \right|$ $(\because C_1\rightarrow C_1-C_2)$
$ = \left| {\begin{array}{{}{c}}4&{{{\left( {{a^x} - {a^{ - x}}} \right)}^2}}&1\\4&{{{\left( {{b^y} - {b^{ - y}}} \right)}^2}}&1\\4&{{{\left( {{c^z} - {c^{ - z}}} \right)}^2}}&1\end{array}} \right|$ $[\because (\alpha+\beta)^2+(\alpha-\beta)^2=4\alpha\beta]$
$ = 4\left| {\begin{array}{{}{c}}1&{{{\left( {{a^x} - {a^{ - x}}} \right)}^2}}&1\\1&{{{\left( {{b^y} - {b^{ - y}}} \right)}^2}}&1\\1&{{{\left( {{c^z} - {c^{ - z}}} \right)}^2}}&1\end{array}} \right|$ $\left( \because {{C_1}\left( {\frac{1}{4}} \right)} \right)$
$=4\times0=0$ $(\because C_1=C_3)$
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