Light through the first polaroid

The incident light is unpolarised with intensity $I_0$.

When unpolarised light passes through a polaroid, its intensity becomes half: $I_1 = \frac{I_0}{2}$

Light through the second polaroid

The second polaroid $B$ is inclined at $45^\circ$ to the first polaroid $A$.

By Malus' law, the transmitted intensity is $I_2 = I_1 \cos^2 \theta$ where $\theta = 45^\circ$.

So, $I_2 = \frac{I_0}{2} \cos^2 45^\circ$

Evaluate $\cos^2 45^\circ$

$\cos 45^\circ = \frac{1}{\sqrt{2}}$ $\cos^2 45^\circ = \frac{1}{2}$

Therefore, $I_2 = \frac{I_0}{2} \cdot \frac{1}{2} = \frac{I_0}{4}$

Match with the options

$\boxed{\frac{I_0}{4}}$

This corresponds to Option C.