Nuking the Mosquito — ridiculously complicated ways to achieve very simple results Here is a toned down example of what I'm looking for:
Integration by solving for the unknown integral of $f(x)=x$:
$$\int x \, dx=x^2-\int x \, dx$$
$$2\int x \, dx=x^2$$
$$\int x \, dx=\frac{x^2}{2}$$
Can anyone think of any more examples?
P.S. This question was inspired by a question on MathOverflow that I found out about here. This question is meant to be more general, accepting things like solving integrals and using complex numbers to evaluate simple problems. 

Do you also accept algebraic propositions? If so:
Finite domains are fields (the standard proof is a one liner: note that multiplication by a nonzero element is injective hence surjective):
Let $A$ be a finite domain. Let $K$ its field of fractions. Certainly $K$ is a finite $A$-module. Consider $K/A \otimes_A K/A$. This is obviously $0$. But for finitely generated $A$ modules $M, N$ it is $\operatorname{supp}(M \otimes_A N) = \operatorname{supp}(M) \cap \operatorname{supp}(N)$. In other words $M \otimes M = 0$ implies $M=0$. Hence in our situation $K=A$.
