1. The problem is to determine whether the function $f(x) = \cos^7(x)$ is odd or even. 2. Recall that a function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain. 3. A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$. 4. We know that $\cos(x)$ is an even function since $\cos(-x) = \cos(x)$. 5. Therefore, for $f(x) = \cos^7(x)$, we have: $$f(-x) = (\cos(-x))^7 = (\cos x)^7 = f(x).$$ 6. Since $f(-x) = f(x)$, the function $f(x) = \cos^7(x)$ is an even function. 7. Final answer: $\cos^7(x)$ is an even function.