2. Given the line $l: y=x-a$ intersects the parabola $C$: $x^{2}=2 p y(p>0)$ at points $M$ and $N$, and the circle passing through $M$ and $N$ intersects the parabola $C$ at two other distinct points $E$ and $F$. Then the cosine of the inclination angle of the line $E F$ is $\qquad$.
2. $-\frac{\sqrt{2}}{2}$.

Let points $M\left(x_{1}, y_{1}\right), N\left(x_{2}, y_{2}\right), E\left(x_{3}, y_{3}\right)$, $F\left(x_{4}, y_{4}\right)$, and the circle passing through points $M, N, E, F$ be
$$
x^{2}+y^{2}+d x+e y+f=0 \text {. }
$$

Then the coordinates of points $M, N, E, F$ satisfy the system of equations
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+d x+e y+f=0, \\
x^{2}=2 p y .
\end{array}\right.
$$

Eliminating $y$ yields
$$
\begin{array}{l}
x^{4}+\left(4 p^{2}+2 p e\right) x^{2}+4 p^{2} d x+4 p^{2} f=0 \\
\Rightarrow x_{1}+x_{2}+x_{3}+x_{4}=0 .
\end{array}
$$

And $k_{M N}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{x_{1}+x_{2}}{2 p}, k_{E F}=\frac{x_{3}+x_{4}}{2 p}$
$$
\Rightarrow k_{M N}+k_{E F}=0
$$
$\Rightarrow$ The inclination angles of lines $M N$ and $E F$ are supplementary.
Given that the inclination angle of $M N$ is $45^{\circ}$, the inclination angle of $E F$ is $135^{\circ}$.
Thus, the cosine value of the inclination angle of line $E F$ is $-\frac{\sqrt{2}}{2}$.