They have similar but very distinct meanings, so context won’t help.
The dictionary says PRSC and PRSECEUT — either outline would work equally well for either word.
The official choice makes sense historically. Prosecute is a legal term, and would have been used much more often, so it got a brief form and persecute is spelled out.
In my life, though, both words are used about equally, and not often enough for me to remember. Even if I write the official version, I’ll probably doubt myself months later when reading it back.
Ideas?