I see. So it's simply phase factors, ##\exp(\mathrm{i} \varphi)##. So what's discussed here is
$$\psi=\psi_1 + \psi_2=R_1 \exp(\mathrm{i} \varphi_1) + R_2 \exp(\mathrm{i} \varphi_2),$$
with ##R_1,R_2>0##. Then of course, with ##\phi=\varphi_2-\varphi_1|##:
$$|\psi|^2=\psi^* \psi = |R_1+R_2 \exp(\mathrm{i} \phi)|^2=[R_1+R_2 \exp(\mathrm{i} \phi)][R_1+R_2 \exp(-\mathrm{i} \phi)]=R_1^2 + R_2^2 +R_1 R_2 [\exp(\mathrm{i} \phi) + \exp(-\mathrm{i} \phi)]=R_1^2+R_2^2 + 2 R_1 R_2 \cos \phi.$$
Since ##R_1=|\psi_1|## and ##R_2=|\psi_2|## that's indeed the equation given in the OP.
The important point is that only relative phases in superposition play a role. Wave functions, which differ only in a phase factor (or "a phasor") represent the same pure quantum state of the electron, i.e., it's not the normalized vectors but the (unit) rays that represent pure states in quantum mechanics.