While developing a authoritative server to control game state and physics, is it worth to also duplicate the physics engine inside the client and do a sanity check every n frames or this is just throwing resources away since most of time the client will be out of sync?

In many cases the client uses physics to extrapolate future positions/collisions and to simulate things that only need to be simulated on the client.  This allows the client to be more resilient to lag, and can help to provide a more detailed client side experience.

Generally speaking the Authority in the server comes from the fact that both sides are running the same simulation concurrently.  The server having the authoritative version.  The client taking steps to remain in sync at all times.

That all being said, you don't have to have physics on the client(depends some on game of course), but I think you lose more than you would probably gain in most scenarios.

Thanks Septopus, yes. I think that keep the client with physics and just rewind in case of disagreement is a better approach. I still need to learn more about this.

If you haven't read these yet:

http://www.gabrielgambetta.com/client-server-game-architecture.html

https://gafferongames.com/post/what_every_programmer_needs_to_know_about_game_networking/

https://gafferongames.com/post/introduction_to_networked_physics/

Best of luck with your project!