Assume, $\frac{x}{1}=\frac{y}{0}=\frac{z}{-1}=\lambda$

So, coordinate of any point on the line is $\left(\lambda ,0,-\lambda \right)$.

As $N$ is the foot of the perpendicular,

$\overrightarrow{PN}·\left(\hat{i}-\hat{k}\right)=0$

$\left(\left(\lambda -1\right)\hat{i}+\left(0-2\right)\hat{j}+\left(-\lambda +1\right)\hat{k}\right)·\left(\hat{i}-\hat{k}\right)=0$

$\Rightarrow \left(\left(\lambda -1\right)\hat{i}+\left(0-2\right)\hat{j}+\left(-\lambda +1\right)\hat{k}\right)·\left(\hat{i}-\hat{k}\right)=0$

$\Rightarrow \left(\left(\lambda -1\right)\times 1+\left(-2\right)\times 0+\left(-\lambda +1\right)\times -1\right)=0$

$\Rightarrow \lambda -1+0+\lambda -1=0$

$\Rightarrow 2\left(\lambda -1\right)=0$

$\Rightarrow \lambda =1$

Hence, the coordinate of $N$$=(1,0,-1)$

Now,

Assume, the coordinate of $Q=\left(\mu ,0,-\mu \right)$

So, $\overrightarrow{PQ}·\left(\hat{i}+\hat{j}+2\hat{k}\right)=0$

$\left(\left(\mu -1\right)\hat{i}+\left(0-2\right)\hat{j}+\left(-\mu +1\right)\hat{k}\right)·\left(\hat{i}+\hat{j}+2\hat{k}\right)=0$

$\left(\mu -1\right)\times 1+\left(0-2\right)\times 1+\left(-\mu +1\right)\times 2=0$

$\mu -1+-2+-2\mu +2=0$

$-\mu -1=0$

$\Rightarrow \mu =-1$

So, the coordinate of $Q=(-1,0,1)$

Hence, $\overrightarrow{PN}=2\hat{j}$ and $\overrightarrow{PQ}=2\hat{i}+2\hat{j}-2\hat{k}$

$\Rightarrow \cos \alpha =\frac{\overrightarrow{PN}·\overrightarrow{PQ}}{\left|\left(\overrightarrow{PN}\right)\left(\overrightarrow{PQ}\right)\right|}$

$=\frac{\left(2\hat{j}\right)·\left(2\hat{i}+2\hat{j}-2\hat{k}\right)}{\left(\sqrt{2^2}\right)\left(\sqrt{2^2+2^2+2^2}\right)}=\frac{1}{\sqrt{3}}$