Answer
$Y=X$
Work Step by Step
Step 1. Identify the given quantities: equation $xy=x+y$, rotation $ \phi=\frac{\pi}{4}$
Step 2. Recall the formula for the transformation:
$x=X\cdot cos\frac{\pi}{4} - Y\cdot sin\frac{\pi}{4}=\frac{\sqrt 2}{2}(X-Y)$,
$y=-X\cdot sin\frac{\pi}{4}+Y\cdot cos\frac{\pi}{4}=\frac{\sqrt 2}{2}(-X+Y)$
Step 3. Plug-in the expressions above to the equation:
$\frac{\sqrt 2}{2}(X-Y)\frac{\sqrt 2}{2}(-X+Y)=\frac{\sqrt 2}{2}(X-Y)+\frac{\sqrt 2}{2}(-X+Y)$
Step 4. Simply the above equation to get $-(X-Y)^2=\sqrt 2(0)=0$ which gives $Y=X$
