$$ABCD$$ is a square. $$E, F$$ are points on $$BC, CD$$ respectively and $$\angle EAF = 45^\circ$$. The value of $$\frac{EF}{BE+DF}$$ is

This is a standard square configuration: rotating triangle $$ABE$$ by $$90^\circ$$ about $$A$$ so that $$AB$$ maps onto $$AD$$ shows that $$EF$$ exactly equals $$BE+DF$$, since the $$45^\circ$$ angle condition makes triangle $$AEF$$ congruent to the rotated triangle. So the ratio $$\frac{EF}{BE+DF}$$ is 1.