You can do this by javascript easily. Register a startup script from server side, when your button is clicked. And that javascript function, will display the div.
Another, if your div is server side, you can add this to updatepanel dynamically, so once your updatepanel gets updated, you div will also be updated. Just you need to display it on button click event
Have a look to the links, to know, how to achieve this
Click here
Click here 2
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