To solve this problem, we need to convert the given Cartesian equation $x^2 - y^2 = 4$ into polar coordinates. The polar coordinate relations between $x, y$ and $r, \theta$ are as follows:

$x = r \cos \theta$ $y = r \sin \theta$

Now, substitute these into the given equation $x^2 - y^2 = 4$:

$(r \cos \theta)^2 - (r \sin \theta)^2 = 4$

Simplifying:

$r^2 \cos^2 \theta - r^2 \sin^2 \theta = 4$

Factoring out $r^2$ on the left side:

$r^2 (\cos^2 \theta - \sin^2 \theta) = 4$

Using the trigonometric identity $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$:

$r^2 \cos 2\theta = 4$

Thus, the equation in polar coordinates is:

$r^2 \cos 2\theta = 4$

This matches option d. Therefore, the correct answer is d.

Would you like any further details or explanations?

Here are five related questions to consider:

1. How would you convert a different equation, such as $x^2 + y^2 = 9$, to polar coordinates?
2. What are other common identities used in converting between Cartesian and polar coordinates?
3. Can you explain the significance of the identity $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$?
4. How would this process change for an equation involving $xy$ terms?
5. What is the geometric interpretation of the polar coordinate equation $r^2 \cos 2\theta = 4$?

Tip: Always look for known trigonometric identities when converting between Cartesian and polar coordinates, as they often simplify the expressions significantly.