An ac is given by equationI=I1cosωt+I2sinωt . The rms value of current is given by:
I1+I22
(I1+I2)22
12(I12+I22)
I12+I222
I=I1cosωt+I2sinωt
(I2)mean=I12cos2ωt¯+I22sin2ωt¯+2I1I2cosωtsinωt¯
=I12×12+I22×12+2I1I2×0
Irms=(I12+I22)1/22