#### Answer

$\theta = 63.4^{\circ}$

#### Work Step by Step

When unpolarized light passes through a polarizing filter, the intensity of the transmitted light is $0.5~I_0$ where $I_0$ is the original intensity. Thus the intensity is $I_1 = 0.5~I_0$ after passing through the first filter.
We can find the angle $\theta$ between the two filters:
$I_2 = I_1~cos^2~\theta$
$0.1~I_0 = 0.5~I_0~cos^2~\theta$
$cos^2~\theta = 0.20$
$cos~\theta = \sqrt{0.20}$
$cos~\theta = 0.447$
$\theta = cos^{-1}~(0.447)$
$\theta = 63.4^{\circ}$