When a polaroid sheet is rotated between two crossed polaroids then the transmitted light intensity will be maximum for a rotation of :

When a polaroid is placed between two crossed polaroids (at 90° to each other), and the middle polaroid makes angle $$\theta$$ with the first:

Intensity after first polaroid: $$I_0/2$$

After middle polaroid: $$\frac{I_0}{2}\cos^2\theta$$

After last polaroid (at $$90° - \theta$$ to middle): $$\frac{I_0}{2}\cos^2\theta \cos^2(90°-\theta) = \frac{I_0}{2}\cos^2\theta\sin^2\theta = \frac{I_0}{8}\sin^2 2\theta$$

Maximum when $$\sin^2 2\theta = 1$$, i.e., $$2\theta = 90°$$, i.e., $$\theta = 45°$$.

The answer is $$45°$$, which corresponds to Option (4).