An alternating current is given by
`I = i_1 cos omegat + i_2 sin omegat`.
The rms current is given by

Explanation

Let us assume that `delta` is an angle such that `tan delta= (i_(2))/(i_(1))` hence we can write the following i:
` sin delta= (i_(2))/(sqrt(i_(1)^(2)+i_(2)^(2)))` and `cos delta= (i_(1))/(sqrt(i_(1)^(2)+i_(2)^(2)))`
Current from the source is given as follows :
`i= i_(1) sin omegat+i_(2) cos omegat `
Dividing and multiplying the above equation with `sqrt(i_(1)^(2)+i_(2)^2)` to get the following :
`i=(sqrt(i_(1)^(2)+i_(2)^(2)))[sinomegat(i_(1))/(sqrt(i_(1)^(2)+i_(2)^(2)))+cosomegat (i_(2))/(sqrt(i_(1)^(2)+i_(2)^(2)))]`
`implies i= (sqrt(i_(1)^(2)+i_(2)^(2)))[sin omegat cosdelta+cosomegatsin delta`
`implies i = (sqrt(i_(1)^(2)+i_(2)^(2))) sin (omegat+delta)`
Hence peak current ` sqrt(i_(1)^(2)+i_(2)^(2))` is flowing in the circuit . In a sinusoidal current rms value is peak value divided by `sqrt(2)` . Hence rms current can be written as follows :
`i_("rms")= sqrt((i_(1)^(2)+i_(2)^(2))/(2))`