Answer
At an angle of $45^{\circ}$, the transmitted intensity is a maximum.
Work Step by Step
(a) Since the light is polarized in the x-direction initially, the intensity of the light after passing through the first polarizer is $I_0~cos^2~\theta$
After passing through the second polarizer, we can find the intensity of the light that is transmitted:
$I = (I_0~cos^2~\theta)~cos^2(90^{\circ}-\theta)$
$I = I_0~cos^2~\theta~sin^2~\theta$
$I = I_0~(sin~\theta~cos~\theta)^2$
$I = I_0~(\frac{1}{2}~sin~2\theta)^2$
$I = \frac{1}{4}~I_0~sin^2~2\theta$
The intensity of the light that is transmitted through the second sheet is $\frac{1}{4}~I_0~sin^2~2\theta$
(Note that $~~sin~2\theta = 2~sin~\theta~cos~\theta$)
(b) The intensity of the transmitted light is a maximum when $sin~2\theta = 1$, which is true when $\theta = 45^{\circ}$. At an angle of $45^{\circ}$, the transmitted intensity is a maximum.
