1. **State the problem:** Find the equation of the tangent line to the curve $y = x^2 + \cos(6x)$ at $x = \pi$. 2. **Formula used:** The equation of the tangent line at $x = a$ is given by: $$y = f(a) + f'(a)(x - a)$$ where $f'(a)$ is the derivative of $f(x)$ evaluated at $x = a$. 3. **Find $f(\pi)$:** $$f(\pi) = (\pi)^2 + \cos(6\pi) = \pi^2 + 1$$ (since $\cos(6\pi) = 1$ because $6\pi$ is a multiple of $2\pi$) 4. **Find the derivative $f'(x)$:** $$f'(x) = \frac{d}{dx}(x^2) + \frac{d}{dx}(\cos(6x)) = 2x - 6\sin(6x)$$ 5. **Evaluate $f'(\pi)$:** $$f'(\pi) = 2\pi - 6\sin(6\pi) = 2\pi - 6 \times 0 = 2\pi$$ (since $\sin(6\pi) = 0$) 6. **Write the tangent line equation:** $$y = f(\pi) + f'(\pi)(x - \pi) = \pi^2 + 1 + 2\pi(x - \pi)$$ 7. **Simplify the equation:** $$y = \pi^2 + 1 + 2\pi x - 2\pi^2 = 2\pi x - \pi^2 + 1$$ **Final answer:** $$y = 2\pi x - \pi^2 + 1$$