Stenli,
<!DOCTYPE html>
<html>
<head>
<title>Untitled</title>
<meta charset="utf-8">
<style type="text/css">
</style>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.12.4/jquery.min.js"></script>
<script>
$(document).ready(function() {
$('#sendTo').on('click', function () {
var insertData = $('#insertArea').val();
var div = $("<div>", {html : insertData});
$('#pastArea').val($(">", div).html().trim());
});
});
</script>
</head>
<body>
<form name="insertForm" id="insertForm">
<div style="width:45%; float: left; margin: auto;">
<textarea name="insertArea" id="insertArea" cols="80" rows="40"><doctype! html>
<html>
<head></head>
<body>
<div></div>
</body>
</html>
</textarea>
</div>
</form>
<br>
<input type="button" name="sendTo" value="Send to" id="sendTo">
<form name="pastForm" id="pastForm">
<div style="width:45%; float: right; margin: auto;">
<textarea name="pastArea" id="pastArea" cols="80" rows="40"></textarea>
</div>
</form>
<br>
</body>
</html>