MySQL ERROR 1045 (28000): Access denied for user ‘root’@’localhost’ (using password: YES)

root@server [~]# mysql

ERROR 1045 (28000): Access denied for user ‘root’@’localhost’ (using password: YES)
root@server [~]#

root@server [/var/lib/mysql/mysql]# ls -lhd /var/lib/mysql
drwxr-x–x 179 mysql mysql 12K Aug 25 01:44 /var/lib/mysql/

root@server [/var/lib/mysql/mysql]# chmod  711 /var/lib/mysql/mysql

root@server1 [/var/lib/mysql/mysql]# ls -lhd /var/lib/mysql/mysql
drwx–x–x 2 mysql mysql 4.0K Aug 25 01:32 /var/lib/mysql/mysql/

root@server1 [/var/lib/mysql/mysql]# service mysql restart
Shutting down MySQL.. [  OK  ]

Starting MySQL                 [  OK  ]

root@server1 [/var/lib/mysql/mysql]# mysql
Welcome to the MySQL monitor.  Commands end with ; or \g. Your MySQL connection id is 1
Server version: 5.0.51a-community MySQL Community Edition (GPL)Type ‘help;’ or ‘\h’ for help. Type ‘\c’ to clear the buffer.

mysql> quit

Bye