The Spain international was recruited for €44 million this summer after firing Diego Simeone's side into the final, where they were eventually beaten by Real Madrid, but the Blues were casualties in the last four, with Costa netting a penalty in a decisive 3-1 second-leg victory.
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"Now I can say I'm really sorry for the part I played in the game we won at Stamford Bridge," Costa told reporters.
"I remember the atmosphere and the passion of the supporters. They stayed right until the end, getting behind their team, even though it was unlikely they could win after we scored two quick goals. I liked that.
"I hope my actions on the pitch and my feet will do the talking. All I can say is I love playing hard and fairly. I want to improve, not only as a player but as an individual, and I believe this is the perfect place for me to do that.
"Joining was an easy decision for me. We have some of the best players in the world, the best manager in the world and it's a club with great history. That's the reason I chose Chelsea and why I want to defend this shirt."