For 400 V & f=50Hz, a 7.5kW motor has cos(fi)=0.8. If we connect this motor to a 400 V and f = 60Hz, what it will be the new cos(fi) ?

The power factor will remain unchanged.

As regard to the cos fi I cannot answer immediately, but regarding to the speed of it, it would be clearly increased. This is the typycal methos to vary speed in motors.

vt=vmxsin(wt+30)

vt is voltage value vm is peak value of voltage T is period for one cycle in seconds  w is angular frequency and30 is the initial value in radians

so,

w=2xpix50=100pi   cos t =0.8/100*cos(100 *pi)  =0.00888 this follows that t=1

R=V/I=53.3

Pmax=sin to the power of2 of omega time =sin2(100*t)=0.017 and for the second thread it is0.021

Pt=It*R this follows It=Pt/R=0.000394 

and in the same flow of equations V=1.0549

and so after these calculation we conclude that

these time discrerte functions after equalized are the same  THUS

Vm*cos(100*pi*t+30)= Vm*cos(120*pi*t+30)

but here comes the benefit of the value of the current which gives values of current voltages

This concluded that 

cos fi =0.25