Final answer:
Achilles’ life changes after he apologizes to Agamemnon by returning to battle with renewed purpose, driven by the loss of Patroclus, and accepts his fate, manifesting themes of heroism and destiny.
Step-by-step explanation:
The question regarding how Achilles' life differs after he apologizes to Agamemnon is rooted in an understanding of The Iliad and Greek mythology. Achilles, although a mighty warrior, is known to have an inevitable vulnerability—his Achilles heel. Before his apology, Achilles is wrathful and refuses to fight, leading to dire consequences for the Greek army. After reconciling with Agamemnon, he returns to battle, motivated by grief over the loss of his friend Patroclus. This change marks a turning point for Achilles, as he embraces his fate and the inevitability of his mortality, illustrating core themes of heroism and tragic destiny that are pervasive in Greek literature and art.