How to show/hide an element on checkbox checked/unchecked states using jQuery?

Attach onchange event to the checkbox:

<input class="coupon_question" type="checkbox" name="coupon_question" value="1" onchange="valueChanged()"/>

<script type="text/javascript">
    function valueChanged()
    {
        if($('.coupon_question').is(":checked"))   
            $(".answer").show();
        else
            $(".answer").hide();
    }
</script>

Try this

$(".answer").hide();
$(".coupon_question").click(function() {
    if($(this).is(":checked")) {
        $(".answer").show(300);
    } else {
        $(".answer").hide(200);
    }
});

FIDDLE