do you need to insure a trailer for towing your car if the car is already insured.
Please clarify what you mean. Is this a trailer that has a car sitting on it, an RV trailer being towed behind a car, or ??? It makes a lot of difference re insurance. Also, are you asking about liability for damage done to others, or repairs to car or trailer itself?
Generally any trailer towed behind another vehicle is covered by the liability insurance on the towing vehicle. In other words, if the trailer causes damage to somebody or something else while underway, the tow vehicle liability insurance covers that damage. It does not, however, pay for repairs to the trailer itself. You would need separate insurance on the trailer for that.
In the case of a car on a utility trailer, being towed by some other vehicle, it gets complex. Damage to the car on the trailer is covered by the cars own collision or comprehensive insurance (if it has it). Damage to the trailer te car is sitting on is not covered by the car insurance or the tow vehicle insurance - it would need its own collision/comp insurance. However, if the trailer broke loose and hit a person or vehicle, the tow vehicle liability covers that. If the car fell off the trailer and hit something, then the car's own liability policy is the primary coverage but the tow vehicle liability is secondary if needed.