1. **Stating the problem:** We want to understand why the expression $\cos(\pi n)$ is equal to $(-1)^n$ for integer values of $n$. 2. **Formula and explanation:** Recall the cosine function values at multiples of $\pi$: $$\cos(\pi n) = \cos(\pi \times n)$$ where $n$ is an integer. 3. **Key property:** The cosine function has the property that: $$\cos(\pi) = -1$$ and $$\cos(0) = 1$$ 4. **Evaluating for integer $n$:** Since $n$ is an integer, the angle $\pi n$ corresponds to $n$ half-turns around the unit circle. The cosine value alternates between 1 and -1 depending on whether $n$ is even or odd. 5. **Using the power of $-1$:** We know that: - If $n$ is even, $(-1)^n = 1$ - If $n$ is odd, $(-1)^n = -1$ 6. **Conclusion:** Therefore, $$\cos(\pi n) = (-1)^n$$ because the cosine of $\pi n$ alternates between 1 and -1 exactly as $(-1)^n$ does. This is a fundamental identity used often in trigonometry and Fourier analysis.