This inequality follows immediately from the inequalities 
$$|e^{ix}-1-ix-(ix)^2/2|\le|x|^3/6$$
and
$$|e^{ix}-1-ix|\le x^2/2$$
for real $x$, as the latter inequality also implies $|e^{ix}-1-ix-(ix)^2/2|\le x^2/2+x^2/2=x^2$. 

In turn, the two displayed inequalities follow immediately from [Taylor's theorem with the integral form of the remainder][1]. 


  [1]: https://en.wikipedia.org/wiki/Taylor%27s_theorem