I don't if this is the simpler solution, but it's a solution
We need to find the coordinates of the 2 points lying on the inclined plane. I don't know if you have those available, but I'm assuming the worst case, which is not.
Let's throw some names:
: is half the angle between the lines (10deg in your example)
: is the angle between the planes (30deg in your example)
: is the point where the 2 lines intercept. I'm assuming this will always be the origin, like in your picture
and [tex]P2[\math]: are the points that define the angle
I'll take these points first lying on the horizontal plane, then rotate them by degrees to find the new coordinates.
They will rotate about the X axis. The rotation matrix for this axis is:
Multiply the point by the matrix and you have the coordinates of the rotated point:
This 2 points are now projected on the horizontal plane. In this particular case x and y coordinates are the same, and z = 0. So, the projected points are:
These 2 points make 2 vector with the origin:
These 2 vector lie on the horizontal plane, now just find the angle between them using the definition of dot product:
Solving this leads to:
Finally, for your example we have
I tried to give you all the steps so you could understand where the answer came from. I hope that was it you needed =)