What are some interesting examples where evaluating an expression assuming its existence is much easier than proving existence?
Simple examples are given by limits of several variables (because we can often find a path along which the limit is trivial). For example, assuming the existence of $$\lim_{(x,y)\to(0,0)}\frac{xy^2+\sin(x)x^2}{x^2+y^2}\qquad\text{and}\qquad \lim_{(x,y,z)\to(0,0,0)} (x^2+y^2+z^2)^{x^2y^2z^2},$$ we obtain $$\lim_{(x,y)\to(0,0)}\underbrace{\frac{xy^2+\sin(x)x^2}{x^2+y^2}}_{:=f(x,y)}=\lim_{y\to 0} f(0,y)=\lim_{y\to 0} 0=0$$ and $$\lim_{(x,y,z)\to(0,0,0)} \underbrace{(x^2+y^2+z^2)^{x^2y^2z^2}}_{:=g(x,y,z)}=\lim_{x\to 0} g(x,0,0)=\lim_{x\to 0} 1=1.$$