A recursive language minus a recursively enumerable language results in a recursive language?
Let $L$ be an r.e. language which is not recursive (for example, $L$ can be the language of descriptions of Turing machines that halt on the empty input). Then $\Sigma^*$ is recursive but $\Sigma^* \setminus L$ is not recursive (exercise).



More generally, when faced with a question like this, try the following two approaches in parallel:


Try to prove the claim.
Try to refute the claim using a counterexample.


The two approaches can even feed each other.