Expectation Values of Angular Momentum Operators

Homework Statement

Show that

< l,m | Lx2 - Ly2 | l,m > = 0

Homework Equations

L2 = Lx2 + Ly2 + Lz2

[ Lx, Ly ] = i [STRIKE]h[/STRIKE] Lz

[ L, Lz ] = i [STRIKE]h[/STRIKE] Lx

[ Lz, Lx ] = i [STRIKE]h[/STRIKE] Ly

The Attempt at a Solution

I tried substituting different commutation values in place of Lx and Ly, but I'm not reducing it any further. I also tried ladder operations, but my professor said they're not needed to solve the problem.

$$\langle l,m \lvert L_x^2 \lvert l,m \rangle = \langle l,m \lvert L_y^2 \lvert l,m \rangle$$