JQuery, radio button getting value

$('#mySelect')
    .find('option')
    .remove()
    .end()
    .append('<option value="whatever">text</option>')
    .val('whatever')
;

Since the question refers to a single element, this code might be more suitable:

// Checks CSS content for display:[none|block], ignores visibility:[true|false]
$(element).is(":visible");

// The same works with hidden
$(element).is(":hidden");

It is the same as twernt's suggestion, but applied to a single element; and it matches the algorithm recommended in the jQuery FAQ.

We use jQuery's is() to check the selected element with another element, selector or any jQuery object. This method traverses along the DOM elements to find a match, which satisfies the passed parameter. It will return true if there is a match, otherwise return false.