2. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, the angle formed by $B C_{1}$ and the section $B B_{1} D_{1} D$ is ( ).
(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{3}$
(D) $\frac{\pi}{2}$
2. A.

As shown in Figure 2, draw $C_{1} O \perp B_{1} D_{1}$ at point $O$, and connect $B O$. Then $\angle C_{1} B O$ is the required angle.
Given $C_{1} O=\frac{1}{2} B C_{1}$,
we know $\angle C_{1} B O=\frac{\pi}{6}$.