13. (20 points) Find the minimum value of the function
$$
\begin{aligned}
f(x)= & \mid \sin x+\cos x+\tan x+\cot x+ \\
& \sec x+\csc x \mid
\end{aligned}
$$

where $\sec x=\frac{1}{\cos x}, \csc x=\frac{1}{\sin x}$.
Three, 13. Let $u=\sin x+\cos x$, then $\sin x \cdot \cos x=\frac{1}{2}\left(u^{2}-1\right)$.
Thus, $\sin x+\cos x+\tan x+\cot x+\sec x+\csc x$ $=u+\frac{2}{u-1}$.
When $u>1$, we have
$$
f(x)=1+u-1+\frac{2}{u-1} \geqslant 1+2 \sqrt{2} .
$$

When $u<1$, we have
$$
f(x)=-1+1-u+\frac{2}{1-u} \geqslant 2 \sqrt{2}-1 \text {, }
$$

When $u=1-\sqrt{2}$, the equality holds.
Therefore, the minimum value of $f(x)$ is $2 \sqrt{2}-1$.