cosh x ≤ e^(x²/2)

Series.
$$\cosh x=\sum_{k\ge0}\frac{x^{2k}}{(2k)!},\qquad e^{x^2/2}=\sum_{k\ge0}\frac{x^{2k}}{2^kk!}.$$

Comparison. For $k\ge1$,
$$(2k)!=k!\cdot(k+1)(k+2)\cdots(2k)\ \ge\ k!\cdot2^k,$$
since each of the $k$ factors $k+1,\dots,2k$ is at least $2$. Hence $\frac1{(2k)!}\le\frac1{2^kk!}$ for all $k$ (equality for $k=0$ and $k=1$).

Conclusion. The inequality holds term by term and $x^{2k}\ge0$, so $\cosh x\le e^{x^2/2}$.

Remark. This bound is the standard "sub-Gaussian" estimate: it shows that a random sign $\pm1$ has moment generating function $\mathbb{E}\,e^{tX}=\cosh t\le e^{t^2/2}$, the key to Hoeffding's inequality.