Which one of the following auto policy coverages would pay for the damages to the insured's car that were caused by hitting a farmer's cattle while driving through the country?

The appropriate coverage in this scenario is Other-than-collision, also known as comprehensive coverage. This type of coverage is specifically designed to address damages to the insured's vehicle that are not a result of a collision with another vehicle or object but rather due to various other incidents. Hitting a farmer's cattle while driving is an example of an incident that falls under this coverage, as it does not involve another car but rather an animal.

Collision coverage typically applies to damage from a collision with another vehicle or object, while liability coverage is intended to cover damages or injuries to another person or their property that the insured may cause in an accident. Medical payments coverage is for medical expenses that the insured or passengers may incur as a result of an accident, not for damages to the vehicle itself. Thus, Other-than-collision is the correct choice for the damages caused in this particular situation.