Spray the rusted areas of the faucet with silicone spray until they're coated. Allow the silicone spray to sit on the surfaces for 10 to 20 minutes.
Scrub the rusted areas with an old toothbrush to removed softened rust stains. Rinse the toothbrush frequently with water. Wipe the faucet with paper towels to remove the rust as it's loosened. Repeat Steps 1 and 2 as needed until enough rust is removed to restore movement to areas of the faucet that were stuck.
Pour 1/8 cup of scouring powder into a bowl. Add 1/8 cup of cream of tartar. Mix enough hydrogen peroxide to form a paste.
Dip the toothbrush in the peroxide paste. Scrub the rusted areas with the paste until coated. Allow the paste to sit on the surface of the faucet for 30 minutes.
Moisten the toothbrush with water. Scrub the paste-coated areas of the faucet to remove the rust stains. Rinse the surface with water to wash away rust and the hydrogen peroxide paste.
Dry the faucet thoroughly with a clean cloth.