Look for a shoot that has 3 to 5 leaf nodes on the stem, and cut under the last leaf node. Keep your cuttings fresh in water while you work. Don't let them dry out.
Use a sharp scissor to snip off the lower leaves, being careful not to damage the leaf node. The roots will grow from the last leaf node.
Fill a clean container with sterile soil. (Tip: Fill to heaping and then firm down soil.) Use a pencil or dibber to make a deep hole. Place your cutting into the hole and carefully push soil towards the stem to bury it. Gently firm down soil around cutting.
Add more cuttings as desired, and water generously. Place into a bright location with indirect sunlight. Too much sunlight will dry out the cuttings too quickly, so shade is good. Water daily to ensure that the cuttings never dry out. You can shower the whole plant, leaves and all, so that the whole plant gets a good soak each day. Ensure that your container has good drainage so that it doesn't drown in water.
In three weeks time, your cuttings will be ready to transplant. Gently tip your container to pour the cuttings out. Separate the cuttings from each other, and be careful not to damage the roots.
See the fine, white roots that have grown out from the leaf node? I have rinsed them out a little in a bucket of water to show you them.
Cosmos have much finer, angel hair-like roots compared to the impatiens and dahlias, huh?
After you replant the cuttings, water well. Water is very important to the success of cuttings. Dried out cuttings simply won't work so be sure to keep cuttings fresh in water after you cut them, and even after you separate them. I like to keep mine floating/submerged in water while I work so that they get a good drink of water before they try to grow roots or get used to a new environment. As long as your container has good drainage, watering every day will be A-okay. However, if your container doesn't have good drainage, you need to watch out for overwatering. (Tip: The soil should be damp, but not soaking.)