Given:
Current flowing through C and L : i₂
Current flowing through R : i₁
Supply AC voltage: v=v_msinωt
Formula used:
We know the equation for LCR circuit

Here, L is the inductance, di/dt is the rate of flow of current, R is the resistance, q is the charge, C is the capacitance and V is the AC voltage.
For first branch (I) consisting resistor R,


i₁ is the current in the first branch and R is the resistance.
For the Second branch (II) consisting of Inductor and Capacitor,

Here, q₂ is the charge due to current i₂.
We know that, A.C value of charge is

q_m is the maximum charge due to maximum current i_m and ϕ is the phase of the circuit. Substituting value of q₂ we get.


If phase is zero then, ϕ =0


Thus current through branch 2: i₂ = dq₂/dt


As i₁ and i₂ have sin and cos terms respectively, we say that both are 90° out of phase. At ϕ =0
Also, total current would be: i=i₁+i₂

Let us use trigonometric property
Let

And


Here A = ((Acosϕ)²+(Asinϕ )²)^1/2


Hence is the expression for total current.
Impedance is given as

V is the voltage and I is the current.


Hence is the expression for impedance of the given circuit.