This inequality has invited lots of incorrect replies, and the correct proofs are surprisingly complicated, involving difficult calculations and estimates. The asker seems to be looking for something that perhaps a (very) clever undergraduate could come up with. Curious to see if anyone here might have any luck
The reference in the Q to e^i0 and the unit circle makes me feel like a fruitful avenue may be to pursue the fact that e^ix is cosx + i sinx and avoid anything to do with critical points. Doubly so when you recall e^i(pi) + 1 = 0 has similarity to the Q expression. e ... circles ... it has to be this path? But a) being a middle aged math newbie (thanks mathacademy.com!) I'm not yet au fait with proofs, and b) no reference to this on the stackexchange page, I am likely off track, but curious what the better avenue is if it isn't that. It almost feels like the Q is asking for a proof of the euler formula. But I am rather naive at this level of math, for now :-)
I don't think those are bad instincts -- especially given that simplicity is desired! So far none of the answers on that page meet the brief, so I wouldn't read much signal from the fact that no one has approached it that way