A completely unpolarized light of intensity ${\mathrm{I}}_0$ is incident on a polarizer, then the emergent ray passes through an analyzer. The intensity of final emergent ray is $\frac{{\mathrm{I}}_0}{8}$. If analyzer is rotated about its own geometrical axis through an angle of ${90}^{\mathrm{o}}$, intensity of final emergent ray will be 

Intensity of light emerging from the polarizer is $\frac{{\mathrm{I}}_0}{2}$ and this light is plane polarized. 

If an angle between the axis of transformation of polarizer and that of analyzer is $\mathrm{\theta }$, then intensity of light coming out is

 $\mathrm{I}=\frac{{\mathrm{I}}_0}{2}{\cos }^2\mathrm{\theta }=\frac{{\mathrm{I}}_0}{8}\Rightarrow \mathrm{\theta }={60}^{\mathrm{o}}$

After the analyzer is rotated, intensity of final emergent light is given by

$\mathrm{I}=\frac{{\mathrm{I}}_0}{2}{\cos }^2\left(\frac{\mathrm{\pi }}{2}\pm \mathrm{\theta }\right)=\frac{{\mathrm{I}}_0}{2}{\cos }^2\left({90}^{\mathrm{o}}\pm {60}^{\mathrm{o}}\right)=\frac{3{\mathrm{I}}_0}{8}$