$ x = rac{\pi}{2} $: $ \cos 2x = \cos \pi = -1 $, $ \cos 4x = \cos 2\pi = 1 $, so $ f(x) = 1 - rac{1}{2}(-1 - 1) = 1 - rac{1}{2}(-2) = 1 + 1 = 2 $. Correct.

["SEO-Optimized Article: Understanding the Identity $ x = \frac{\pi}{2} $ and $ \cos 2x = \cos \pi = -1 $, $ \cos 4x = \cos 2\pi = 1 $ — A Key Step in Trigonometric Function Analysis", "---", "Exploring the Mathematical Identity at $ x = \frac{\pi}{2} $: A Deep Dive into Cosine Behavior", "The equation $ x = \frac{\pi}{2} $ serves as a fundamental entry point in trigonometric reasoning, especially when analyzing the behavior of cosine functions at key angular values. Substituting $ x = \frac{\pi}{2} $ into fundamental cosine identities reveals powerful symmetries and functional relationships that are essential in both pure mathematics and applied fields like physics and engineering.", "### Evaluating $ \cos 2x $ When $ x = \frac{\pi}{2} $", "Start with the double-angle identity for cosine:\n$$\n\cos 2x = \cos\left(2 \cdot \frac{\pi}{2}\right) = \cos \pi\n$$\nSince $ \cos \pi = -1 $, we confirm:\n$$\n\cos 2x = -1\n$$", "### Evaluating $ \cos 4x $ at $ x = \frac{\pi}{2} $", "Next, compute the second angle for the fourth-order cosine:\n$$\n\cos 4x = \cos\left(4 \cdot \frac{\pi}{2}\right) = \cos 2\pi\n$$\nAs $ \cos 2\pi = 1 $, we find:\n$$\n\cos 4x = 1\n$$", "### Deriving the Function $ f(x) = 1 - \frac{1}{2}(-1 - 1) $", "To connect these values meaningfully, consider a function $ f(x) $ defined as:\n$$\nf(x) = 1 - \frac{1}{2}(\cos 2x - \cos 4x)\n$$\nSubstituting $ x = \frac{\pi}{2} $:\n$$\nf\left(\frac{\pi}{2}\right) = 1 - \frac{1}{2}(-1 - 1) = 1 - \frac{1}{2}(-2) = 1 + 1 = 2\n$$\nThis demonstrates how alternating cosine terms interact within algebraic expressions — a technique often used in signal processing and Fourier analysis to capture wave behavior and differences.", "### Significance and Applications", "The identity $ \cos 2x = -1 $, $ \cos 4x = 1 $ at $ x = \frac{\pi}{2} $ highlights cosine’s periodic nature with predictable turning points at multiples of $ \pi $. These key angles serve as benchmarks for evaluating trigonometric expressions and constructing function graphs. Understanding such identities strengthens foundational knowledge crucial for solving differential equations, analyzing oscillations, and modeling periodic phenomena.", "### Conclusion", "The evaluation $ x = \frac{\pi}{2} $ reveals elegant symmetry in cosine functions: $ \cos 2x = -1 $ and $ \cos 4x = 1 $ — a powerful illustration of trigonometric periodicity. The function $ f(x) = 1 - \frac{1}{2}(\cos 2x - \cos 4x) $ exemplifies how these values combine to produce concrete numerical results, reinforcing the value of direct substitution and identity application in mathematical reasoning.", "Whether used in high school trigonometry or advanced engineering applications, recognizing these key identities — like $ \cos \frac{\pi}{2} = 0 $ — enables deeper insight and precise computation.", "---", "Keywords:\n$ x = \frac{\pi}{2} $, $ \cos 2x $, $ \cos 4x $, trigonometric identities, periodic functions, mathematical reasoning, signal processing, Fourier analysis, mathematics education", "Meta Description:\nExplore the key trigonometric identity $ x = \frac{\pi}{2} $, where $ \cos 2x = -1 $ and $ \cos 4x = 1 $, and how these values apply in evaluating functions like $ f(x) = 1 - \frac{1}{2}(\cos 2x - \cos 4x) $. Ideal for students and math enthusiasts.", "---", "Transform your understanding of periodic functions and build mastery in trigonometry with precision and clarity."]