The sum of the co-efficients of all odd degree terms in the expansion of ${(x+\sqrt{x^3-1})}^5+{(x-\sqrt{x^3-1})}^5, (x>1)$

$\begin{array}{rcl}\sqrt{{\mathrm{x}}^3-1}& =& \mathrm{y}\\ & & {(\mathrm{x}+\mathrm{y})}^5+{(\mathrm{x}-\mathrm{y})}^5 \\ & =& 2 \left[{}^5{\mathrm{C}}_0 {\mathrm{x}}^5+{}^5{\mathrm{C}}_2 {\mathrm{x}}^3 {\mathrm{y}}^2+{}^5{\mathrm{C}}_4 \mathrm{x} {\mathrm{y}}^4\right]\end{array}$

$\begin{array}{rcl} & =& 2 \left[{}^5{\mathrm{C}}_0 {\mathrm{x}}^5+{}^5{\mathrm{C}}_2 {\mathrm{x}}^3 ({\mathrm{x}}^3-1)+{}^5{\mathrm{C}}_4 \mathrm{x} {({\mathrm{x}}^3-1)}^2\right]\\ & =& 2 \left[{\mathrm{x}}^5+10\left({\mathrm{x}}^6-{\mathrm{x}}^3\right)+5\left({\mathrm{x}}^7+\mathrm{x}-2{\mathrm{x}}^4\right)\right]\end{array}$

Sum of odd degree terms

$=2 \left[1-10+5+5\right]=2$