By mistake, I have created a duplicate database in the phpMyAdmin page of cPanel. I want to delete this database, but I am not able to find any delete button in the UI.
You can delete the database in Cpanel.
In Cpanel go to databases and there you will see all created databases and you can delete this database.
http://docs.cpanel.net/twiki/bin/view/AllDocumentation/CpanelDocs/MySQLDatabases#Delete a Database
or other wise on live server if user have privileges you can run command in sql tab of phpmyadmin.
drop database databasename;