mimeType = file.toURL().openConnection().getContentType();
does not work, since this use of URL leaves a file locked, so that, for example, it is undeletable.
However, you have this:
and also the following, which has the advantage of going beyond mere use of file extension, and takes a peek at content
InputStream is = new BufferedInputStream(new FileInputStream(file));
mimeType = URLConnection.guessContentTypeFromStream(is);
However, as suggested by the comment above, the built-in table of mime-types is quite limited, not including, for example, MSWord and PDF. So, if you want to generalize, you'll need to go beyond the built-in libraries, using, e.g., Mime-Util (which is a great library, using both file extension and content).