Two particles are executing simple harmonic motion at an instant of time 't' their displacements are  y1=acos(wt) and y2=asin(wt) . Then the phase difference between y1  and y2  is

y1=acos(wt)            =asinwt+π/2, here phase angle φ1=wt+π/2               y2=asin(wt), here phase angle φ2=wt         phase difference =Δφ=φ1-φ2                                              Δφ=wt+π/2-wt=π2 radians                                             Δφ=90∘       (in degrees)