If x = a + 1/a and y = a - 1/a, then the value of x4 + y4 - 2x2y2 is?

Answer: Option C

$$\begin{array}{rl}& x=a+\frac{1}{a}\\ & y=a-\frac{1}{a}\\ & \therefore \left(x+y\right)=a+\frac{1}{a}+a-\frac{1}{a}=2a\\ & \therefore \left(x-y\right)=a+\frac{1}{a}-a+\frac{1}{a}=\frac{2}{a}\\ & \therefore x^4\text{ + }{\text{y}}^4-2x^2{y}^2\\ & ={\left(x^2-y^2\right)}^2\\ & ={\left[\left(x+y\right)\left(x-y\right)\right]}^2\\ & ={\left(2a\times \frac{2}{a}\right)}^2\\ & ={\left(4\right)}^2\\ & =16\end{array}$$