If $$x = a + \frac{1}{a}   and   y = a - \frac{1}{a}$$ then $$\sqrt{x^4 + y^4 - 2x^2y^2}$$ is equal to:

As per the question,

$$x = a + \frac{1}{a}$$ and  $$y = a - \frac{1}{a}$$

Squaring both side,

$$x^2 = (a + \frac{1}{a})^2=a^2+(\dfrac{1}{a})^2+2$$

Similarly

$$y^2 = (a + \frac{1}{a})^2=a^2+(\dfrac{1}{a})^2-2$$

Now, $$\sqrt{x^4 + y^4 - 2x^2y^2}=\sqrt{(x^2-y^2)^2}$$---------(i)

Substituting the values in the equation (i)

$$\sqrt{x^4 + y^4 - 2x^2y^2}=\sqrt{(x^2-y^2)^2}=\sqrt{(a^2+(\dfrac{1}{a})^2+2-a^2-(\dfrac{1}{a})^2+2)^2}=4$$