Can Taylor Expansion Really "Prove" the Euler Formula?

Euler's formula, the most well-known formula in complex analysis, provide us an easy approach to compute the exponential of imaginary numbers. Almost all physics mathematics and engineering mathematics teachers "prove" Euler formula with Taylor exansion. However, this approach confused me for a long time.
My calculus teacher is a strict teacher. She required usto write every tiny details in our proof. To meet her expectations, I often tried my best to memorize every word in the theorems. When I was taking calculus, teacher said that Taylor expansion can only be employed on real functions. The behavior of utilizing Taylor expansion on a complex function is not a method I can accept. However, Euler is one of the greatest mathematicians, was the proof he gave wrong?
I ultimately got the answer to this problem after reading a book about complex analysis. Though the Taylor expansion indeed can be employed on complex functions, the proof of this claim requires Euler formula. That is to say, Euler's proof indeed wrong. Nevertheless, this mistake is not due to him.
I noticed that the base of mathematical analysis, the modern definition of limit, that is, the \(\epsilon-\delta\) definition was give by Cauchy, a great mathematician in the 19th century. However, Euler was a mathematician in the 18th century. In other words, Euler did not have the \(\epsilon-\delta\) definition. Therefore, the complex analysis was not complete during Euler's period. Furthermore, it is not strange that there are some problems in Euler's proof.
Due to the completeness of mathematical analysis, Euler's formula can be strictly proved now. However, the proof given by Euler is still well-known and widely taught to students outside of mathematics. This experience really taught me not to believe teachers and textbooks blindly. I try my best to doubt everything carefully and often remind myself of the well-known saying by Mencius (孟子)「盡信書不如無書」 (To believe all in books is not as good as having no books).