To tame cosmo:

It is a passive time with Chitan. A light pet is required.

Then, it is basic red light green light. He will flash a white/green light, you then sprint to him and feed him. He also can flash a red light, DONT move, as he will simply toss webs at you. Once he switches back to green/white, run back to feed him

Repeat until tamed. Also, note that once tamed, the web shooting ability appears as an item in your inventory.