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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.

Find an equation equivalent to x^2 - y^2 = 4 in polar coordinates.

Learn how to convert the Cartesian equation x^2 - y^2 = 4 into polar coordinates. This step-by-step solution covers key concepts such as the relation between Cartesian and polar coordinates and the use of trigonometric identities. Suitable for high school students in Grades 10-12.

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