\(\sin^2 2x\) の範囲は \([0, 1]\) です。したがって、\(\frac{1}{2} \sin^2 2x\) の最大値は \(\frac{1}{2}\) であり、\(1 - \frac{1}{2} \sin^2 2x\) の最小値は、\(\sin^2 2x = 1\) のときに達成されます:

["Understanding the Range of (\sin^2(2x)) and Key Values", "The expression (\sin^2(2x)) is a fundamental trigonometric function that plays a crucial role in many areas of mathematics, physics, and engineering. One key fact every learner should master is that the range of (\sin^2(2x)) is precisely ([0, 1]). This means the squared sine of (2x) always varies between 0 and 1, inclusive.", "### What Does This Mean?", "Since (\sin(2x)) oscillates between (-1) and (1), squaring it ensures the result is never negative and capped at 1. Thus:", "[\n0 \leq \sin^2(2x) \leq 1\n]", "This equality defines the bounds for trigonometric calculations involving this function.", "### Exploring Key Transformed Expressions", "Now consider the two related expressions:", "[\n\frac{1}{2} \sin^2(2x) \quad \ ext{and} \quad 1 - \frac{1}{2} \sin^2(2x)\n]", "Because (\sin^2(2x)) lies in ([0, 1]), let’s analyze the transformed values:", "1. Maximum value of (\frac{1}{2} \sin^2(2x)):", "When (\sin^2(2x) = 1),\n[\n\frac{1}{2} \cdot 1 = \frac{1}{2}\n]\nSo, the maximum of (\frac{1}{2} \sin^2(2x)) is (\frac{1}{2}).", "2. Minimum value of (1 - \frac{1}{2} \sin^2(2x)):", "The minimum occurs when (\sin^2(2x)) reaches its maximum of 1:\n[\n1 - \frac{1}{2} \cdot 1 = \frac{1}{2}\n]", "Alternatively, when (\sin^2(2x) = 0),\n[\n1 - \frac{1}{2} \cdot 0 = 1\n]\nSo, the expression (1 - \frac{1}{2} \sin^2(2x)) varies between (\frac{1}{2}) and 1, reaching minimum ( \frac{1}{2} ) when ( \sin^2(2x) = 1 ).", "### Practical Implications", "Understanding these bounds helps simplify trigonometric optimization problems, model wave behaviors, and solve differential equations. Knowing when expressions attain minima and maxima enables accurate predictions and efficient mathematical modeling.", "### Summary", "- The range of (\sin^2(2x)) is ([0, 1]).\n- (\frac{1}{2} \sin^2(2x)) has a maximum value of (\frac{1}{2}).\n- (1 - \frac{1}{2} \sin^2(2x)) reaches its minimum value of (\frac{1}{2}) when (\sin^2(2x) = 1).", "Mastering these relationships enhances clarity in trigonometry and lays a strong foundation for advanced study."]