The 2002 Mercury Cougar has two primary fuse/relay box locations: the engine compartment and the passenger side compartment. Your starter relay resides in the first of the two. As the starter relay, like most primary automobile relays, is a "plug-in" or "cartridge-style" relay, removal and replacement take very little time.
Disconnect the battery. Keep in mind that if you don't disconnect the battery, you run the risk of an accidental electrical surge that could harm you or damage your Mercury Cougar.
Remove the cover of your fuse/relay box--also referred to as the power distribution box in your owner's manual. Unplug the starter relay located in the top left corner of the power distribution box.
Pull the relay and inspect it and its socket for damage, dirt or corrosion. Clean or repair the relay, socket or box fuse/relay housing as needed and check the socket pins for damage, such as bent areas or irregular pin height.
Replace your Mercury Cougar's starter relay with a new relay of equal amperage. Keep in mind that using wires or a higher capacity relay as a temporary relay "fix" can cause electrical surges that could damage wiring or start a fire. In addition, make certain the relay fits securely in its socket as current running through a loose relay can potentially melt the relay's outer casing, surrounding relay covers and the fuse/relay box cover.
Reconnect your battery, after replacing the fuse/relay box cover and start your Mercury Cougar to confirm the new relay works.
