These results are for observations made by a dip circle. Usually we use a dip circle like this:
1.First allow needle to rotate in horizontal plane. This ensure that the needle is now in the magnetic meridian. In this situation, only H is applying a torque on the needle that can move it. Torque applied by V doesn't move the needle because this torque tends to rotate needle in vertical plane but we have restricted needles rotation to only horizontal plane.
2. Now, allow the needle rotate in vertical plane. Now, V will also apply a torque and the needle will stop when it aligns with B at that point, where B is net magnetic field.
In the above pic, magnetic meridian is the grey coloured plane. The dotted plane represents some other random plane at an angle \alpha with magnetic meridian.
So if the needle were first allowed to rotate on horizontal plane and then vertical, it would show the true dip angle \delta
Now, if we omit step 1 and just keep the needle in some other arbitrary plane at an angle \alpha with magnetic meridian. (here we don't allow the needle to rotate in horizontal plane)
In this new plane, V is same (clear from figure). H has a component H\cos \alpha perpendicular to V and in this new plane. So the needle will now stop along the resultant of H\cos \alpha and V and hence showing a different dip \delta_1
What happens to H \sin \alpha? It still exists but the torque it applies on the needle try to move it in a different plane so it is of no consequence to us. We are allowing only those torques to act on the needle which will tend to rotate it in the new plane.