Determine the maximum value of \((\sin x + \cos x)^2 + (\sin x - \cos x)^2\).
["# Determine the Maximum Value of ((\sin x + \cos x)^2 + (\sin x - \cos x)^2)", "Understanding trigonometric expressions is fundamental in mathematics, physics, and engineering. One common challenge is determining the maximum value of combinations like ((\sin x + \cos x)^2 + (\sin x - \cos x)^2). This article explores how to simplify this expression, determine its maximum value, and explains the significance in real-world applications.", "## Simplifying the Expression", "Start with the given expression:", "[\n(\sin x + \cos x)^2 + (\sin x - \cos x)^2\n]", "Use the algebraic identity ((a + b)^2 + (a - b)^2 = 2a^2 + 2b^2):", "Let (a = \sin x) and (b = \cos x), so:", "[\n(\sin x + \cos x)^2 + (\sin x - \cos x)^2 = 2\sin^2 x + 2\cos^2 x\n]", "Factor out the 2:", "[\n= 2(\sin^2 x + \cos^2 x)\n]", "Using the Pythagorean identity (\sin^2 x + \cos^2 x = 1):", "[\n= 2 \cdot 1 = 2\n]", "## Finding the Maximum Value", "Since the expression simplifies to a constant:", "[\n(\sin x + \cos x)^2 + (\sin x - \cos x)^2 = 2\n]", "This means the maximum value (and indeed the only possible value) is 2, regardless of the value of (x).", "## Why Is This Important?", "This result is powerful because it shows that regardless of (x), this combination of squares always sums to 2. This has implications in signal processing, wave analysis, and optimization problems where such trigonometric identities simplify computation.", "## Step-by-Step Summary:", "1. Expand both squared terms:\n ((\sin x + \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x)\n ((\sin x - \cos x)^2 = \sin^2 x - 2\sin x \cos x + \cos^2 x)", "2. Add the expansions:\n ((\sin x + \cos x)^2 + (\sin x - \cos x)^2 = (\sin^2 x + \cos^2 x + 2\sin x \cos x) + (\sin^2 x + \cos^2 x - 2\sin x \cos x))", "3. Combine like terms and use identity:\n (= 2(\sin^2 x + \cos^2 x) = 2 \cdot 1 = 2)", "Since (\sin^2 x + \cos^2 x = 1) always holds, the expression is constant.", "## Conclusion", "The maximum value of ((\sin x + \cos x)^2 + (\sin x - \cos x)^2) is 2, achieved for all real (x). This elegant identity highlights how trigonometric expressions stabilize under simplification, making them useful tools in mathematical modeling and analytical problem-solving.", "For further exploration, verify this result numerically by plotting the expression or evaluating it across various (x) values — consistent results will confirm the theoretical conclusion.", "---", "Keywords: ((\sin x + \cos x)^2 + (\sin x - \cos x)^2), maximum value, trigonometric identity, mathematical simplification, (\sin^2 x + \cos^2 x = 1), mathematical analysis, optimization, constant value."]
