1. **State the problem:** Simplify the expression $$1 + \cos(4x) - \cos(6x) - \frac{\cos(10x) + \cos(2x)}{2}$$ using the product-to-sum formula for the last term. 2. **Recall the product-to-sum formula:** $$\cos A \cos B = \frac{\cos(A+B) + \cos(A-B)}{2}$$ This formula allows us to rewrite products of cosines as sums of cosines. 3. **Given:** $$\cos(6x) \cos(4x) = \frac{\cos(10x) + \cos(2x)}{2}$$ This matches the product-to-sum formula with $A=6x$ and $B=4x$. 4. **Substitute back into the expression:** $$1 + \cos(4x) - \cos(6x) - \cos(6x) \cos(4x) = 1 + \cos(4x) - \cos(6x) - \frac{\cos(10x) + \cos(2x)}{2}$$ 5. **Rewrite the expression with a common denominator of 2:** $$= \frac{2}{2} + \frac{2\cos(4x)}{2} - \frac{2\cos(6x)}{2} - \frac{\cos(10x) + \cos(2x)}{2}$$ 6. **Combine all terms over 2:** $$= \frac{2 + 2\cos(4x) - 2\cos(6x) - \cos(10x) - \cos(2x)}{2}$$ 7. **Final simplified form:** $$\boxed{\frac{2 + 2\cos(4x) - 2\cos(6x) - \cos(10x) - \cos(2x)}{2}}$$ This is the simplified expression after applying the product-to-sum formula and combining terms. **Explanation:** We used the product-to-sum formula to rewrite the product of cosines as a sum, then substituted it back and combined all terms carefully with a common denominator to simplify the expression.