Short answer

God (﷾) did not need to wait, because He never needed a death in order to forgive. The Qur'an says that Adam repented after eating from the tree, and God (﷾) accepted his repentance. No sacrifice was needed, and no one else paid for his sin. On the Christian account, God (﷾) knew from Adam's day that people would sin, yet the remedy came thousands of years later. The Bible itself shows God (﷾) forgiving people who turned back to Him, long before the cross Christians speak of. And God (﷾), who is almighty, can forgive by His will alone. In Islam, a believer's hardship also has a purpose. This life is a test, and God (﷾) wipes away sins through the hardships a believer bears.

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