9. (16 points) As shown in Figure 1, let the base of the pyramid $E-ABCD$ be a rhombus, and $\angle ABC=60^{\circ}, AB=EC=2$, $AE=BE=\sqrt{2}$.
(1) Prove: Plane $EAB \perp$ Plane $ABCD$;
(2) Find the cosine value of the dihedral angle $A-EC-D$.

保留了原文的换行和格式。
(1) Take the midpoint $O$ of $AB$, and connect $EO$, $CO$. Since $AE=EB=\sqrt{2}, AB=2$, we know that $\triangle AEB$ is an isosceles right triangle.
Thus, $EO \perp AB, EO=1$.
Also, $AB=BC, \angle ABC=60^{\circ}$, so $\triangle ACB$ is an equilateral triangle.
Therefore, $CO=\sqrt{3}$.
Given $EC=2$,
$EC^{2}=EO^{2}+CO^{2}$, and $EO \perp CO$.
Hence, $EO \perp$ plane $ABCD$.
Since $EO$ lies on plane $EAB$, 
plane $EAB \perp$ plane $ABCD$.
(2) As shown in Figure 2, take the midpoint $O$ of $AB$ as the origin, and the lines $OC$, $OB$, and $OE$ as the $x$-axis, $y$-axis, and $z$-axis, respectively, to establish a spatial rectangular coordinate system.
Then $A(0,-1,0), C(\sqrt{3}, 0,0)$
$$
\begin{array}{l}
D(\sqrt{3},-2,0), E(0,0,1), \\
\overrightarrow{AC}=(\sqrt{3}, 1,0), \overrightarrow{EC}=(\sqrt{3}, 0,-1), \\
\overrightarrow{DC}=(0,2,0) .
\end{array}
$$

Let $\boldsymbol{n}=(x, y, z)$ be the normal vector of plane $DCE$. Then
$$
\left\{\begin{array}{l}
\vec{EC} \cdot \boldsymbol{n} = 0, \\
\vec{DC} \cdot \boldsymbol{n} = 0
\end{array} \Rightarrow \left\{\begin{array}{l}
\sqrt{3} x - z = 0, \\
2 y = 0 .
\end{array}\right.\right.
$$

Let $x=1$. Hence $z=\sqrt{3}$.
Therefore, $\boldsymbol{n}=(1,0, \sqrt{3})$.
Let $\boldsymbol{m}=(a, b, c)$ be the normal vector of plane $EAC$. Then
$$
\left\{\begin{array}{l}
\vec{AC} \cdot \boldsymbol{m} = 0, \\
\vec{EC} \cdot \boldsymbol{m} = 0
\end{array} \Rightarrow \left\{\begin{array}{l}
\sqrt{3} a + b = 0, \\
\sqrt{3} a - c = 0 .
\end{array}\right.\right.
$$

Let $a=1$. Hence
$$
b=-\sqrt{3}, c=\sqrt{3}, \boldsymbol{m}=(1,-\sqrt{3}, \sqrt{3}) .
$$

Then $\cos \langle\boldsymbol{m}, \boldsymbol{n}\rangle=\frac{\boldsymbol{m} \cdot \boldsymbol{n}}{|\boldsymbol{m}||\boldsymbol{n}|}=\frac{2 \sqrt{7}}{7}$.
Therefore, the cosine value of the dihedral angle $A-EC-D$ is $\frac{2 \sqrt{7}}{7}$.