Has any club won both in a same season? 
Seven clubs have won the FA Cup as part of a League and Cup double, namely Preston North End (1889), Aston Villa (1897), Tottenham Hotspur (1961), Arsenal (1971, 1998, 2002), Liverpool (1986), Manchester United (1994, 1996, 1999) and Chelsea (2010). In 1993, Arsenal became the first side to win both the FA Cup and the League Cup in the same season when they beat Sheffield Wednesday 2–1 in both finals. Liverpool (in 2001) and Chelsea (in 2007) have since repeated this feat. In 2012, Chelsea accomplished a different cup double consisting of the FA Cup and the 2012 Champions League. In 1998–99, Manchester United added the 1999 Champions League title to their league and cup double to complete a unique Treble. Two years later, in 2000–01, Liverpool won the FA Cup, League Cup and UEFA Cup to complete a cup treble. An English Treble has never been achieved.