Wow, I'm a Korean native and I'm reading Korean letters but I can't understand. Hahahaha! 털다 emphasizes the action of shaking or beating something. 떨다 emphasizes the result of the action above. In other words, you have an old coat, which has dust on it. So you shake or beat the coat, about which we say 코트(coat)를 털었다. As a result of it, that is, because you beat the coat, dust on it was shaken off, about which we say 먼지(dust)를 떨었다. I hope it'll help. Hahaha Have a nice day/night!