Why did Macduff fight with Macbeth and do you think Macbeth behaved bravely at the moment of his death?

In his final battle with MacDuff, Macbeth fights recklessly at first, but ends with giving in to defeat, seeing that his destiny after all is to be vanquished and killed. In the second prophecy of the witches, Macbeth is told that he will not fall until Great Birnam Wood comes to Dunsinane Hill. This happens as MacDuff’s troops hide behind cut-off branches, making it appear that the woods are marching to war. The witches also prophesied that Macbeth could not be killed by anyone who was born of woman. Macbeth takes this to believe that his death in battle will be impossible, since all human beings are born of woman. He tells this prophecy to MacDuff during their fighting, but MacDuff replies that he was born by Caesarean section (“from his mother’s womb/Untimely ripped” Act 5 Scene 8). At this point, Macbeth realizes that he has misinterpreted the prophecies (or else been tricked by the witches into misinterpretation). He loses heart and is soon killed by MacDuff. Bravery thus is not a trait that he exhibits at this point, but rather foolhardiness and arrogance, believing that he is undefeatable.