How is the component of the velocity along the tooth equal to Vsinα for both the meshing gears?
Making the relative sliding velocity Vs = V1sinα + V1sinα
The pitch line velocities (tangent velocity at the pitch diameter) are the same for both gears in mesh. If they weren't you would have the tooth of one gear traveling through the tooth of the other gear. At the point of contact the pitch line velocities and the forces exerted are equal, thus you have no change in power.
As for the derivation of your equations, I'm not sure what exactly α is in regards to.