This is not done in ASP but in JavaScript.
However, it's fairly simple:
Code:
<input type="radio" name="R1" value="Radio1" onclick="document.getElementById('textArea').style.display='none';" checked />Hide Textarea<br />
<input type="radio" name="R1" value="Radio2" onclick="document.getElementById('textArea').style.display='';" />Show Textarea<br />
<textarea id="textArea" rows="3" cols="10" name="myTextarea" style="display: none;"></textarea>