To solve the equation cos(2x)sin(x) = 1, it is essential to recognize that both cos(2x) and sin(x) can only reach a maximum value of 1. This leads to the conditions where cos(2x) must equal 1 and sin(x) must equal 1, or both must equal -1, but these conditions cannot be satisfied simultaneously. The discussion suggests graphing cos(2x) and sin(x) over the interval [0, 2π] to visualize potential intersections. Additionally, transforming the equation into a cubic form by substituting sin(x) with u can help identify real roots. Ultimately, finding the solutions requires careful analysis of these transformations and graphical representations.