if

We know that $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$

Now,

$\tan ^{-1} x+\tan ^{-1} y=\frac{\pi}{4}$

$\Rightarrow \tan ^{-1}\left(\frac{x+y}{1-x y}\right)=\frac{\pi}{4}$

$\Rightarrow \frac{x+y}{1-x y}=\tan \frac{\pi}{4}$

$\Rightarrow \frac{x+y}{1-x y}=1$

$\Rightarrow x+y=1-x y$

$\Rightarrow x+y+x y=1$

$\therefore x+y+x y=1$