Given:
f(x) = x^2 - 1/x^2

To Find:
f(x) * f(1/x)

Solution:

To find the value of f(x) * f(1/x), let's first find the values of f(x) and f(1/x) separately.

Finding f(x):
Given f(x) = x^2 - 1/x^2

We can rewrite f(x) as:
f(x) = (x^2 * x^2 - 1) / x^2

Simplifying the numerator, we have:
f(x) = (x^4 - 1) / x^2

Now, let's factorize the numerator:
f(x) = [(x^2)^2 - 1] / x^2

Using the difference of squares formula, we can write:
f(x) = [(x^2 + 1)(x^2 - 1)] / x^2

Further simplifying, we have:
f(x) = [(x^2 + 1)(x + 1)(x - 1)] / x^2

Finding f(1/x):
To find f(1/x), we substitute x with 1/x in the original function f(x).

f(1/x) = (1/x)^2 - 1/(1/x)^2

Simplifying, we have:
f(1/x) = 1/x^2 - x^2

Calculating f(x) * f(1/x):
Now, let's find the value of f(x) * f(1/x) by multiplying the two functions.

f(x) * f(1/x) = [(x^2 + 1)(x + 1)(x - 1)] / x^2 * (1/x^2 - x^2)

Multiplying the numerators and denominators, we have:
f(x) * f(1/x) = [(x^2 + 1)(x + 1)(x - 1)(1/x^2 - x^2)] / x^4

Expanding the numerator, we get:
f(x) * f(1/x) = [(x^2 + 1)(x + 1)(x - 1)(1 - x^4)] / x^4

Further simplifying, we have:
f(x) * f(1/x) = [(x^2 + 1)(x + 1)(x - 1)(1 - x^2)(1 + x^2)] / x^4

Conclusion:
The value of f(x) * f(1/x) is [(x^2 + 1)(x + 1)(x - 1)(1 - x^2)(1 + x^2)] / x^4.