1. Stating the problem: Solve the equation $$\cos(x) - x \sin(x) = 0$$ for $x$. 2. Rearrange the equation: $$\cos(x) = x \sin(x)$$ 3. Divide both sides by $\cos(x)$, assuming $\cos(x) \neq 0$: $$1 = x \tan(x)$$ which gives $$x \tan(x) = 1$$ 4. We need to find $x$ such that $$x \tan(x) = 1$$. 5. This transcendental equation cannot be solved exactly by elementary methods, so we consider approximate solutions. 6. Check for obvious solutions: - At $x=0$, $x \tan(x) = 0$ which is not equal to 1. - For small positive $x$, $\tan(x) \approx x$, so $x \tan(x) \approx x^2$ which is less than 1 for small $x$. 7. By numerical methods or graphing, we find the first positive root near $x \approx 0.86$. 8. Also consider negative values: $x \tan(x)$ is odd function, so negative root near $x \approx -0.86$. 9. Conclusion: The solutions to the original equation are approximately $$x \approx \pm 0.86$$. These can be found more precisely using numerical solvers.