Four polaroids are placed at an angle θ with the optic axis of the...

Intensity of the incident unpolarised light is $I$.

When unpolarised light passes through the first polaroid, the transmitted intensity becomes half of the initial intensity. Thus, the intensity after the first polaroid is:
$I_1 = \dfrac{I}{2}$

According to Malus's law, when polarised light of intensity $I_0$ passes through a polaroid placed at an angle $\theta$ with the plane of polarisation, the transmitted intensity is $I_0 \cos^2\theta$.

Intensity after passing through the second polaroid:
$I_2 = I_1 \cos^2\theta = \dfrac{I}{2} \cos^2\theta$

Intensity after passing through the third polaroid:
$I_3 = I_2 \cos^2\theta = \dfrac{I}{2} \cos^4\theta$

Intensity after passing through the fourth polaroid:
$I_4 = I_3 \cos^2\theta = \dfrac{I}{2} \cos^6\theta$

Given that $\cos\theta = X$, substituting this into the expression gives:
$I_4 = \dfrac{I X^6}{2}$

Answer: $\dfrac{IX^6}{2}$