Display tags from specific category in select form

Hello,

I am looking to display all tags in select form that are from posts that have been assigned to a specific category.

I am using the following code to generate every tag in a select form.

<div>
  <?php
  echo "<select onChange=\"document.location.href=this.options[this.selectedIndex].value;\">";
	echo "<option>By product</option>\n";
	foreach (get_tags() as $tag)
	{
	  echo "<option value=\"";
	  echo get_tag_link($tag->term_id);
	  echo "\">".$tag->name."</option>\n";
	}
          echo "</select>"; ?>
</div>

Can someone point me in the direction on how I can just display the tags that are from all posts assigned to the videos category?

Any help much appreciated