1. **State the problem:** Simplify the expression $\cos 2x \cos x + \sin 2x \sin x$. 2. **Recall the formula:** The cosine addition formula states: $$\cos(a - b) = \cos a \cos b + \sin a \sin b$$ 3. **Apply the formula:** Here, let $a = 2x$ and $b = x$, so: $$\cos 2x \cos x + \sin 2x \sin x = \cos(2x - x)$$ 4. **Simplify the argument:** $$\cos(2x - x) = \cos x$$ 5. **Final answer:** $$\cos 2x \cos x + \sin 2x \sin x = \cos x$$ This shows the original expression simplifies neatly to $\cos x$ using the cosine difference identity.