quantum mechanics - Rotation operator

The problem is this: Consider the operator $e^{-\frac{i\pi L_y}{2\hbar}}$. If you apply it to an eigenstate of $L^2$ and $L_x$ with $l = 1$, prove that the resulting state is an eigenstate of $L_z$.

The idea is this:

$$L_z\left ( e^{-\frac{i\pi L_y}{2\hbar}}Y_{1,m} \right )=-i\hbar \frac{\partial }{\partial \varphi }\left ( e^{-\frac{i\pi L_y}{2\hbar}}Y_{1,m} \right )=...=2\hbar ^2\left (e^{-\frac{i\pi L_y} {2\hbar}}Y_{1,m} \right ).$$

But I don't know how to move from one equation to the other. I would appreciate any help. Thank you.