If $x,y,z \in R$, then the value of $\left| {\begin{array}{llllllllllllllllllll}{{{\left( {{2^x} + {2^{ - x}}} \right)}^2}}&{{{\left( {{2^x} - {2^{ - x}}} \right)}^2}}&1\\{{{\left( {{3^x} + {3^{ - x}}} \right)}^2}}&{{{\left( {{3^x} - {3^{ - x}}} \right)}^2}}&1\\{{{\left( {{4^x} + {4^{ - x}}} \right)}^2}}&{{{\left( {{4^x} - {4^{ - x}}} \right)}^2}}&1\end{array}} \right|$ is……..

We have
$\left| {\begin{array}{cccccccccccccccccccc}{{{\left( {{2^x} + {2^{ - x}}} \right)}^2}}&{{{\left( {{2^x} - {2^{ - x}}} \right)}^2}}&1\\{{{\left( {{3^x} + {3^{ - x}}} \right)}^2}}&{{{\left( {{3^x} - {3^{ - x}}} \right)}^2}}&1\\{{{\left( {{4^x} + {4^{ - x}}} \right)}^2}}&{{{\left( {{4^x} - {4^{ - x}}} \right)}^2}}&1\end{array}} \right|$

$= \left| {\begin{array}{llllllllllllllllllll}{\left( {2 \cdot {2^x}} \right)\left( {2 \cdot {2^{ - x}}} \right)}&{{{\left( {{2^x} - {2^{ - x}}} \right)}^2}}&1\\{\left( {2 \cdot {3^x}} \right)\left( {2 \cdot {3^{ - x}}} \right)}&{{{\left( {{3^x} - {3^{ - x}}} \right)}^2}}&1\\{\left( {2 \cdot {4^x}} \right)\left( {2 \cdot {4^{ - x}}} \right)}&{{{\left( {{4^x} - {4^{ - x}}} \right)}^2}}&1\end{array}} \right|\quad$

$= \left| {\begin{array}{cccccccccccccccccccc}4&{{{\left( {{2^x} - {2^{ - x}}} \right)}^2}}&1\\4&{{{\left( {{3^x} - {3^{ - x}}} \right)}^2}}&1\\4&{{{\left( {{4^x} - {4^{ - x}}} \right)}^2}}&1\end{array}} \right| = 0$.

[ since$,{C_1}$ and ${C_3}$ are proportional to each other]