Great question. I don’t have a visual diagram or intuition for tan(a+b) yet, but I think using the dome-wall-ceiling diagram you might be able to construct it (I’d have to play around a bit). Great idea for a follow-up though.

The hypotenuse of the blue triangle is one. That makes any point along it measurable as a percentage value. As Kalid has said before, cos and sin are just percentages.
So we can imagine the length of the base of the red triangle of distance cos (b) as a percentage value.

Now the hypotenuse of the triangle with angle (a) can also be considered a way of measuring points along the vertical distance, sin (a). If you, for example, draw a horizontal line connecting the mid way point of the hypotenuse across to the vertical line sin (a), that gives you the distance 50% of sin (a). If you connect a line between the point of distance cos (b) and the vertical line sin (a), that gives you cos (b) of sin(a), or cos(b)sin(a). So we see that the distance sin(a) reduces to cos (b) sin (a) according to this calculation when brought ‘in’ to match up with that point along the hypotenuse.

Now we can do the same trick with calculating the next term in the equation, which as we know will turn out to be cos (a) sin (b).

We know that the red triangle has been tilted up by an angle of (a). That means as Kalid says that we can no longer use the full value of sin (b).

Instead we can mentally construct a new triangle on its side, at ninety degrees from the horizontal, which is attached to the right side of the red triangle. Since it is attached to the red triangle which is askew from the horizontal by angle (a), it is askew from the vertical by that same angle. The hypotenuse of that triangle adjoins the right side of the red triangle. If we were to draw this third triangle in full, the hypotenuse would extend to a distance one and the length of its ‘base’ or vertical distance would be cos (a).

Hence bringing this length ‘in’ to connect with the distance ‘sin (b)’ provides the value cos (a) sin (b).