**Solution:**

To find the angle of rotation of axes, we need to compare the given equation xy = 0 with the transformed equation X^2 - Y^2 = 0. By comparing the two equations, we can determine how the original x-y coordinate system has been transformed.

Let's analyze the given equation xy = 0:
- This equation represents a pair of intersecting lines, x = 0 and y = 0, which are the x-axis and y-axis respectively.
- The x-axis and y-axis intersect at the origin (0, 0).
- The lines x = 0 and y = 0 are perpendicular to each other, forming a 90-degree angle between them.

Now, let's analyze the transformed equation X^2 - Y^2 = 0:
- This equation represents a pair of intersecting lines, X = Y and X = -Y.
- The lines X = Y and X = -Y intersect at the origin (0, 0).
- The lines X = Y and X = -Y are diagonals of a square in the new coordinate system.

To determine the angle of rotation, we need to find the angle between the original x-axis and the new X-axis. Since the new coordinate system is a rotated version of the original system, this angle of rotation will be equal to the angle between the original x-axis and the line X = Y.

Let's find the angle between the original x-axis and the line X = Y:
- The line X = Y can be written as Y = X.
- This equation represents a line with a slope of 1, which means it is inclined at a 45-degree angle with respect to the x-axis.

Therefore, the angle of rotation of axes is 45 degrees.

In summary, the original x-y coordinate system has been rotated by an angle of 45 degrees to obtain the transformed coordinate system represented by the equation X^2 - Y^2 = 0.

Ans) given xy=0 (eq1)
x= Xcos theta - Ysin theta
y= Xsin theta + Ycos theta
substitute in (eq1)
[Xcos theta - Ysin theta][ Xsin theta +Ycos theta]=0
X^2sintheta costheta + XY cos^2 theta - XYsin^ 2 theta -Y^2 sintheta costheta =0
now
sintheta costheta {X^2 - Y^2 } + XY {cos^2 theta - sin^2 theta } =0

from question {X^2 - Y^2 }=0
so ,
0 + XY { cos2theta } =0
cos2theta = 0
cos2theta = cos π/2
theta = cos π/ 4 ( that is 45 degrees)