When are some instances where someone has used two branches of math that people thought were unrelated to make a new discovery, prove a long-standing conjecture, or something similar?
One example I think of is when Andrew Wiles proved Fermat's Last Theorem by looking at its connections with elliptic curves, something that people thought was completely unrelated.
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This is definitely going to be a much less impressive result than the other answers will provide, but I still think it's very cool:
Consider the space $S$ of Riemannian metrics on some (high enough dimension) compact manifold $M$, and think about the diameter functional $D: S\rightarrow\mathbb{R}$. Clearly $D$ has no local minima - given a Riemannian metric on $M$, I can shrink it to get one of strictly smaller diameter. However, this process makes the curvature of the metric more extreme, so we can de-trivialize the question by restricting to the subspace of $S$, $S'$, consisting of Riemannian metrics with sectional curvature in $[-1, 1]$ (say).
Now it's conceivable that there could be local minima; however, there are no obvious tools for finding them. $D$ is continuous, so we'd be set if $S'$ were compact, but it's very much not. And there don't seem to be any easy ways around this.
Weinberger and Nabutovsky proved that, in fact, $D$ has infinitely many local minima in $S'$ . . . using computability theory! And to the best of my knowledge, no proof avoiding computability theory is known.
The most famous is probably using groups to study fields using Galois Theory. And in the process providing beautiful solutions to classical problems, such as solving equations using algebraic operations, angle trisection, construction of regular polygons. Even Fermat's Last Theorem is proved by making use of Galois Theory.
Some other nice connections, are those provided by the Langlands program.