Above are the question and correct solution. However I would like to know what mistake did I make in my own unfinished solution:
It's symmetric so we can take $0<x<45^{\circ} \implies 0<\tan(x)<1$. The triangle can't exist if and only if $$\cos^2(x)>\sin^2(x)+\sin(x)\cos(x)$$ $$\iff \tan^2(x)+\tan(x)-1 < 0$$ $$\iff \frac{-1-\sqrt{5}}{2}< \tan(x) < \frac{-1+\sqrt{5}}{2}$$