Unpolarized light of intensity I is incident on a system of two polarizers, A followed by B. The intensity of emergent light is I/2. If a third polarizer C is placed between A and B, the intensity of emergent light is reduced to I/3. The angle between the polarizers A and C is θ. Then cosθ = ?

Let initial intensity be I.
Intensity of the beam after passing through A is I1 = I/2
given that intensity after B is I2 = I/2. Then the angle between A and B is zero.
A polarizer C is introduced between A and C then by Malus law
after B Ib = I2cos²θ
and after C Ic = Ibcos²θ
so given Ic = I/3
from here solving all the three equations
I/3 = (I/2)cos⁴θ
cos⁴θ = 2/3
cosθ = (2/3)^(1/4)