Unpolarized light is incident on a polaroid. Intensity of transmitted light is $I_0$. Two such polaroids are placed at ${90}^0$ to each other and $\left(N-1\right)$ more polaroids are inserted between them such that the angle between axes of any two adjacent polaroids is same. Intensity of transmitted light is $I=I_0{\left(\cos \frac{\pi }{2N}\right)}^{KN}.$ find the value of $K$.

Let $I_{in}$ be the intensity of input unpolarized light
Intensity of light after crossing the first polaroid, $I_0=\frac{I_{in}}{2}$
In the second case, after the light crosses the first polarizer, it passes through $N$ more polaroids. While crossing each polaroid, its intensity decreases by a factor of ${\cos }^2\theta$. Final transmitted intensity,
$I=\frac{I_{in}}{2}{\left({\cos }^2\theta \right)}^N=I_0{\left(\cos \theta \right)}^{2N}$

Also, $N\theta =\frac{\mathrm{\pi }}{2}$    $\theta =\frac{\mathrm{\pi }}{2N}$

$I=I_0{\left(\cos \frac{\mathrm{\pi }}{2N}\right)}^{2N}$