Euler’s formula (well, one of Euler’s formulas) states that The proof I am most familiar with is based on the Maclaurin series for , and .
I ran across a different, simpler proof recently on Math Stack Exchange, posted by user02138. It doesn’t give any insight into why the formula is true, but it is very short.
Let . Then Simplifying, we have The fact that implies that for some constant C. Since and thus
Some of the other answers to “How to prove Euler’s formula” are interesting and worth perusing as well.