I tried to prove Euler's formula using two different methods.
I would appreciate if anyone would kindly check if I have made any mistakes with the proves.
proofofeulersformula.pdf
I tried to prove Euler's formula using two different methods.
I would appreciate if anyone would kindly check if I have made any mistakes with the proves.
proofofeulersformula.pdf
Your second method starts with $\displaystyle e^{i\theta}= cos(\theta)+ i sin(\theta)$. convert to $\displaystyle 1= \frac{e^{i\theta}}{cos(\theta)+ i sin(\theta)}$ and show that the derivative of both sides is 0.
That is not logically valid. First, you start by asserting what you want to show. That can be used in what is sometimes called "synthetic proof"- assert what you want to show then reduce to an obviously true statement- provided every step is "invertible". But the derivative is NOT invertible. The fact that f'(x)= g'(x) does NOT imply that f(x)= g(x).