I recently learned that $\cos{\theta} = \frac{e^{i\theta} + e^{-i\theta}}{2}$ and $\sin{\theta} = \frac{e^{i\theta} - e^{-i\theta}}{2}$ Based on this, I managed to "prove" that: $$e^{i\theta} = e^{-i\theta}$$
Since $e^{i\theta} = \cos{\theta} + i\sin{\theta}$, we can substitute the above two identities to get: $$e^{i\theta} = \frac{e^{i\theta} + e^{-i\theta}}{2} + i\frac{e^{i\theta} - e^{-i\theta}}{2}$$ Simplifying, I get $$(1-i)e^{i\theta} = (1-i)e^{-i\theta}$$ which implies $$e^{i\theta} = e^{-i\theta}$$ for all real $\theta$. Obviously, this isn't true in general, but I'm having a hard time seeing what's wrong. Can someone please point out the flaw in the above "proof"?