\(I_{1}=\frac{I_{0}}{2}\)

The intensity of transmitted light through \(P_{3}\)

\({I_2} = {I_1}{\cos ^2}{45^o} = \) \(\frac{{{I_0}}}{2}{\left( {\frac{1}{{\sqrt 2 }}} \right)^2}\) \( = \frac{{{I_0}}}{2} \cdot \frac{1}{2} = \frac{{{I_0}}}{4}\)

Angle between polaroids \(P_{3}\) and \(P_{2}=\left(90^{\circ}-45^{\circ}\right)\) \(=45^{\circ}\)

\(\therefore \quad\) Intensity of transmitted light through \(P_{2}\)

\({I_3} = {I_2}{\cos ^2}{45^o} = \) \(\frac{{{I_o}}}{4}{\left( {\frac{1}{{\sqrt 2 }}} \right)^2} = \frac{{{I_o}}}{8}\)