Tachometer is not working?

Looking at the cluster it appears that the fuel gauge and tachometer both use the same power and ground. The difference is the tach gets an RPM signal from the PCM. Probably the easiest test would be to plug a scan tool or code reader that can display live data into it. Does it display the RPMs? If yes you know the signal side works to the PCM. Depending on the vehicle it may do a sweep test on initial power on (turn the key on, do the rest of the gauges move or move to zero? Did the tachometer? Beyond that there isn't much that can test it without a bench full of test gear to simulate the signals.