If $\alpha and\beta$ be the coefficients of $x^{4}and{x}^2$ respectively in the expansion of ${(x+\sqrt{x^2-1})}^6+{(x-\sqrt{x^2-1})}^6\text{,}$ then 

$$\begin{gathered} 2[6_{c_0}.x^6+6_{c_2}.x^4(x^2-1)+6_{c_4}{x}^2{(x^2-1)}^2+6_{c_6}{(x^2-1)}^3] \\ =2[1.x^6-15x^4(1-x^2)+15x^2{(1-x^2)}^2-{(1-x^2)}^3] \end{gathered}$$

$\alpha =x^4$ coefficient $=2[-15+15(-2)-\left(3\right)]=-96$

$\beta =x^2$ coefficient $=2[15\left(1\right)-(-3)]=36$