A line passes through the origin and makes an angle of $\pi /4$ with the line $x-y+1=0$, then

Given, equation of line

$x-y+1=0$

Slope of line (i), m = 1 

The equation of lines making an angle $\frac{\pi }{4}$with the line (i) and 

passing through origin is 

$y-0=\frac{1\pm \tan \frac{\pi }{4}}{1\mp 1\times \tan \frac{\pi }{4}}(x-0)$

$\Rightarrow y=\frac{1+1}{1-1}\left(x\right)$

and $\begin{array}{r}y=\frac{1-1}{1+1}\left(x\right)\\ x=0\text{ and }y=0\end{array}$