Even though the real Richard Nixon could play the piano, the real Richard Nixon never learned how to read music. During the part of the film when Richard Nixon is playing his piano sheet music is visible on his piano. It is unlikely that Richard Nixon would have had sheet music on his piano, since he didn't read music.
President Nixon presses the record button on his cassette tape recorder and begins recording, but a few moments later realizes that there is no cassette tape in the recorder. Cassette tape recorders have a trip bar inside the cassette compartment that make it impossible for the user to press the record button if no cassette is in the recorder.
As President Nixon is speaking, a closeup of the sound meter on the recorder shows that the needle of the meter does not move.