Some problems turn out to be easier to solve in a domain which is the dual of that where they were originally formulated... I don't recall offhand any examples that I feel familiar enough with to discuss here

One of the most famous dualities is the Fourier transform (and the family of similar transforms). Inter alia, this allows one to easily convolve two functions, which is often valuable in the applied sciences and engineering (or just doing a Gaussian blur in Photoshop).