How to download a youtube video using the youtube's API?
Downloading Youtube videos is against their Terms of Service, so their API's will not support that.
Page linked above refers to Youtube ToS that states:
You shall not download any Content unless you see a “download” or similar link displayed by YouTube on the Service for that Content.
Check out Python API for YouTube, it downloads videos or can just get the direct URL to the video:https://pythonhosted.org/Pafy/
There is obviously no api-side option, but you can simply use youtube-dl and call it via subprocess inside your python script, which is way easier/stable than using on standalone youtube-downloaders.