trigonometry - Understanding Euler's Identity

I would like to understand one specific moment in Euler's Identity, namely

$$e^{j\theta}=\cos(\theta)+j\sin(\theta)$$

where $j=\sqrt{-1}$. We also know that

$$e^{j2(\pi)}=\cos(2\pi)+j\sin(2\pi)$$

but $\sin(2\pi)=0$ and $\cos(2\pi)=1$, and $1=e^{0}$, so we get that $e^{j2(\pi)}=e^{0}$.

But we get that $j2\pi=0$ which means that $j=0$, but on the other hand $j=\sqrt{-1}$. I want to ask one question: why is it allowed to use such symbols in identity, which finally may cause some strange equality?

Answer

The mistake is that $\exp: \mathbb C \to \mathbb C$ is no longer a bijection. Thus
$$e^{2\pi i} = e^{0} \not\Rightarrow 2\pi i = 0$$
In general
$$\exp(z) = \exp(z+2\pi i) \qquad \forall\ z\in\mathbb C$$
because of the periodicity of $\sin$ and $\cos$ and the definition
$$\exp(z) = \underbrace{\exp(\Re z)}_{\exp: \mathbb R\to\mathbb R} \cdot (\cos(\Im z) + i\sin(\Im z))$$