In the proof of the Erdös-Kac theorem in section $III.4$ of $\mathit{Introduction\ to\ Analytic\ and\ Probabilistic\ Number\ Theory}$ by Gérald Tenenbaum, we need to bound the characteristic function $$\varphi_N(\tau) = \frac{1}{N} \sum_{n \leq N} \exp \left\{\frac{i \tau}{\sqrt{\log \log N}}\left(\omega(n) - \log\log N\right) \right\},$$ for $\tau \in \mathbb R,$ where $\omega(n)$ counts the number of distinct prime divisors of $n$.
The given result for $\lvert \tau \rvert \leq T = \sqrt{\log \log N}$ is that $\varphi_N(\tau) \ll e^{-2 \tau ^2 / \pi^2}$. The only steps shown in the proof are to let $t = \tau / T$ and note that $\cos t - 1 \leq -2t^2 / \pi ^2$ for $\lvert t \rvert \leq 1$.
I have spent quite a bit of time trying to see how this follows. Trying to bound term-wise runs into the issue that we just have a complex exponential, so we get the trivial bound of $1$. Otherwise, using the hint, we have that $$\mathfrak{R} \left(\exp\left(\frac{i \tau}{T}\right)\right) = \cos\left(\tau/T\right) \leq 1 - \frac{2 \tau^2}{\pi^2 T^2} \leq \exp \left( - \frac{2\tau^2}{\pi^2 T^2}\right),$$
which would give us the right bound after raising both sides to the $T^2$ power. However, I am unsure how to get a good bound on the imaginary part of the expression.
In the original paper that this proof of the Erdös-Kac theorem is adapted from (A Rényi and P Turán. On a theorem of Erdös-Kac. $\mathit{Acta\ arithmetica}$, 4(1):71–84, 1958), I cannot see this bound being used.
It suffices to notice that when $t=\tau/T$ and $T=\sqrt{\log\log N}$, there is
\begin{aligned} |(\log N)^{e^{it}-1}| &=(\log N)^{\cos t-1}\le(\log N)^{-2t^2/\pi^2} \\ &=e^{-2(\tau/T)^2/\pi^2\cdot T^2}=e^{-2\tau^2/\pi^2}. \end{aligned}