To simplify this problem, let's break it down:

The original equation is: $x^2 - y^2 = a^2$ We need to find the angle of rotation of the axes such that this equation reduces to the form $xy = c^2$.

To do this, we need to apply the formula for the angle of rotation $\theta$ of the coordinate axes. The general transformation equations for the rotation of the axes by an angle $\theta$ are:

$x = x' \cos \theta - y' \sin \theta$ $y = x' \sin \theta + y' \cos \theta$

Substitute these into the equation $x^2 - y^2 = a^2$, and simplify the resulting equation in terms of $x'$ and $y'$. Then, compare it with the equation $x'y' = c^2$ to find the correct angle of rotation $\theta$ and the value of $c^2$.

Let me know if you need further detailed steps, or if you'd like help with the next part!

Would you like to proceed with the calculations?