1. The problem is to evaluate and understand the expression $e^{i\tau} = 1$. 2. Recall Euler's formula: $$e^{i\theta} = \cos(\theta) + i\sin(\theta)$$ where $\theta$ is a real number. 3. Here, $\tau$ is often defined as $2\pi$, so substituting $\tau = 2\pi$ gives: $$e^{i\tau} = e^{i2\pi} = \cos(2\pi) + i\sin(2\pi)$$ 4. Evaluate the trigonometric functions: $$\cos(2\pi) = 1$$ $$\sin(2\pi) = 0$$ 5. Therefore: $$e^{i\tau} = 1 + i \cdot 0 = 1$$ 6. This shows that $e^{i\tau} = 1$ is true because the complex exponential completes one full rotation around the unit circle in the complex plane. Final answer: $$e^{i\tau} = 1$$