1. Given the function
$$
f(x)=\frac{1}{4}\left(x^{2}+\ln x\right) \text {. }
$$

When $x \in\left(0, \frac{\pi}{4}\right)$, the size relationship between $\mathrm{e}^{\cos 2 x}$ and $\tan x$ is
$$
\text { - 1. } \mathrm{e}^{\cos 2 x}>\tan x \text {. }
$$

Obviously, $f(x)=x^{2}+\ln x$ is an increasing function on $(0,+\infty)$.
Since when $x \in\left(0, \frac{\pi}{4}\right)$, $\cos x>\sin x$, therefore,
$$
\begin{array}{l}
f(\cos x)>f(\sin x) \\
\Rightarrow \cos ^{2} x+\ln \cos x>\sin ^{2} x+\ln \sin x \\
\Rightarrow \cos 2 x>\ln \tan x \\
\Rightarrow \mathrm{e}^{\cos 2 x}>\tan x .
\end{array}
$$