This is to complement the inequality 
$$|(1+it)^n-1|\ge b_1(n,t):=|e^{int}-1|,\tag{10}\label{10}$$
proved by [Terry Tao][1] for real $t$ and natural $n$, by the following inequality:
$$|(1+it)^n-1|\ge b_2(n,t):=(1+t^2)^{n/2}-1\ge b_3(n,t):=nt^2/2 \tag{20}\label{20}$$
for real $t$ and real $n\ge0$. 

For any fixed real $t\ne0$, the lower bound $b_2(n,t)$ on $|(1+it)^n-1|$ grows exponentially in $n$, whereas the lower bound $b_1(n,t)$ remains bounded by $2$. More generally, the lower bound $b_2(n,t)$ (and even the lower bound $b_3(n,t)$) will be better than $b_1(n,t)$ if $nt^2$ is large enough (say if $nt^2>4$). 

To prove \eqref{2}, just note that 
$$|(1+it)^n-1|^2=1 - 2 c (1 + t^2)^{n/2} + (1 + t^2)^n\ge b_2(n,t)^2,$$
where $c:=\cos(n\arctan t)\le1$. 

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Here are the graphs $\Big\{\Big(t,\dfrac{|(1+it)^n-1|}{b_1(n,t)}\Big)\colon0<t<1,b_1(n,t)\ge b_2(n,t)\}$ (gold) and $\Big\{\Big(t,\dfrac{|(1+it)^n-1|}{b_2(n,t)}\Big)\colon0<t<1,b_2(n,t)\ge b_1(n,t)\}$ (blue) for $n=100$: 

[![enter image description here][2]][2]


  [1]: https://mathoverflow.net/a/455620/36721
  [2]: https://i.sstatic.net/Bv6l0.png