1. The problem states the famous Euler's formula: $e^{i\pi} = -1$. 2. Euler's formula relates complex exponentials to trigonometric functions: $$e^{ix} = \cos x + i \sin x$$ where $i$ is the imaginary unit and $x$ is a real number. 3. Substitute $x = \pi$ into Euler's formula: $$e^{i\pi} = \cos \pi + i \sin \pi$$. 4. Evaluate the trigonometric functions: $\cos \pi = -1$ and $\sin \pi = 0$. 5. Therefore, $$e^{i\pi} = -1 + i \cdot 0 = -1$$. 6. This shows the beautiful identity connecting exponential and trigonometric functions, known as Euler's identity.