Originally Posted by
FBinNY
Actually it's a bit more complicated than that because wind resistance isn't linear. When you combine the crosswind with the bicycle's forward speed you get an apparent wind speed higher than either. Since wind drag is proportional to the square of wind speed, when you calculate the drag at the higher speed, then divide it back into the two vectors, the forward drag component is higher than it would have been in still air.
So it's not at all an illusion that riding across the wind is harder than riding in still air.
I respectfully disagree with your analysis. Drag exists only in opposition to motion. Since all of the motion is in the forward direction, in the case of a cross wind, only the self generated head wind contributes to drag. As others have explained you still have to expend extra energy fighting a cross wind even though it doesn’t increase drag.