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

Answer: Option C

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