If a- b- c - 0 and x- y- z - R- then the value of the determinant ax - a-x2 ax - a-x2 1 by - b-y2 by - b-y2 1 cz - c-z2 cz - c-z2 1 is -

#160; (a^x + a^ x)^2 amp; (a^x a^ x)^2 amp; 1 (b^y + b^ y)^2 amp; (b^y b^ y)^2 amp; 1 (c^z + c^ z)^2 amp; (c^z c^ z)^2 ...

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Properties of Determinants

If a, b, c ...

Question

If $a, b, c > 0$ and $x, y, z$ $\in$ R, then the value of the determinant $∣ ∣ ∣ ∣ ∣ \begin{matrix} (a^{x} + a^{- x})^{2} & (a^{x} - a^{- x})^{2} & 1 (b^{y} + b^{- y})^{2} & (b^{y} - b^{- y})^{2} & 1 (c^{z} + c^{- z})^{2} & (c^{z} - c^{- z})^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$ is ................

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a, b, c > 0

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∣ ∣ ∣ ∣ ∣ \begin{matrix} {(a^{x} + a^{- x})}^{2} & {(a^{x} - a^{- x})}^{2} & 1 {(b^{y} + b^{- y})}^{2} & {(b^{y} - b^{- y})}^{2} & 1 {(c^{z} + c^{- z})}^{2} & {(c^{z} - c^{- z})}^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

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∣ ∣ ∣ ∣ ∣ \begin{matrix} (a^{x} + a^{- x})^{2} & (a^{x} - a^{- x})^{2} & 1 (b^{y} + b^{- y})^{2} & (b^{y} - b^{- y})^{2} & 1 (c^{z} + c^{- z})^{2} & (c^{z} - c^{- z})^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

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∣ ∣ ∣ ∣ ∣ \begin{matrix} {(a^{x} + a^{- x})}^{2} & {(a^{x} - a^{- x})}^{2} & 1 {(b^{y} + b^{- y})}^{2} & {(b^{y} - b^{- y})}^{2} & 1 {(c^{z} + c^{- z})}^{2} & {(c^{z} + c^{- z})}^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ \begin{matrix} (a^{x} + a^{- x})^{2} & (a^{x} - a^{- x})^{2} & 1 (b^{x} + b^{- x})^{2} & (b^{x} - b^{- x})^{2} & 1 (c^{x} + c^{- x})^{2} & (c^{x} - c^{- x})^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

x \neq a

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∣ ∣ ∣ ∣ \begin{matrix} a & y & z x & b & z x & y & c \end{matrix} ∣ ∣ ∣ ∣ = 0

\frac{x + a}{x - a} + \frac{y + b}{y - b} + \frac{z + c}{z - c}

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