In order to remove a faucet handle for repair or replacement, you have to remove the faucet cap. These plastic caps are usually disguised as the temperature indicators on the top of the faucet. Removing the faucet cap gives you access to the faucet handle screw, which allows the faucet handle to be removed. There are a number of different approaches you can try when trying to loosen a stuck faucet cap. If one doesn't work, move on to the next one or try a combination of these methods if necessary.
Clean off the area around the stuck faucet cap with the bathroom cleaner and rag. This will help clear up any mould or mildew that might be causing the cap to stick.
Pour 1/2 cup of vinegar into the cup. Insert the toothbrush in the cup and get the bristles wet. Brush the vinegar onto the top of the stuck faucet cap. Allow it to sit for about 10 minutes. This will help break up any calcium deposits that may be causing the cap to stick.
Spray the faucet cap liberally with WD-40 or a similar penetrating lubricant. Allow the lubricant time to soak in according to the manufacturer's instructions.
Insert the edge of a sharp knife underneath the faucet cap. Run the knife around the perimeter of the cap. Gently pry the cap upward with the knife.
Insert the edge of a flat-bladed screwdriver underneath the faucet cap as you did with the knife. Pry upward.
Attach a plastic cutting tool to a rotary tool. Put on protective eye wear. Cut around the perimeter of the faucet cap until the cap comes loose.
Use the rotary tool only as a last resort, as this will damage the faucet cap beyond repair.