Let us simplify the expression \((\sin x + \cos x)^2 + (\sin x - \cos x)^2\).
["# Simplifying ((\sin x + \cos x)^2 + (\sin x - \cos x)^2): A Step-by-Step Guide", "In mathematics, simplifying complex expressions is a fundamental skill. One particularly elegant trigonometric identity often encountered is:", "[\n(\sin x + \cos x)^2 + (\sin x - \cos x)^2\n]", "While it appears involved at first glance, the expression simplifies beautifully using basic algebraic and trigonometric identities. This article walks you through the step-by-step simplification, explaining the key concepts and showing how even complex-looking formulas can be made clear and concise.", "---", "## Step 1: Expand Each Square Expression", "We begin by expanding both squared terms using the formula ((a \pm b)^2 = a^2 \pm 2ab + b^2):", "[\n(\sin x + \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x\n]", "[\n(\sin x - \cos x)^2 = \sin^2 x - 2\sin x \cos x + \cos^2 x\n]", "Add both expressions together:", "[\n(\sin x + \cos x)^2 + (\sin x - \cos x)^2 = (\sin^2 x + 2\sin x \cos x + \cos^2 x) + (\sin^2 x - 2\sin x \cos x + \cos^2 x)\n]", "---", "## Step 2: Combine Like Terms", "Now combine the terms:", "[\n= \sin^2 x + \cos^2 x + 2\sin x \cos x + \sin^2 x + \cos^2 x - 2\sin x \cos x\n]", "Observe that (+2\sin x \cos x) and (-2\sin x \cos x) cancel each other:", "[\n= (\sin^2 x + \sin^2 x) + (\cos^2 x + \cos^2 x) + (2\sin x \cos x - 2\sin x \cos x)\n]", "[\n= 2\sin^2 x + 2\cos^2 x + 0\n]", "[\n= 2(\sin^2 x + \cos^2 x)\n]", "---", "## Step 3: Apply the Pythagorean Identity", "This is a well-known identity in trigonometry:", "[\n\sin^2 x + \cos^2 x = 1\n]", "So substitute this identity:", "[\n2(\sin^2 x + \cos^2 x) = 2 \ imes 1 = 2\n]", "---", "## Final Result", "Thus, the simplified form of the expression is:", "[\n(\sin x + \cos x)^2 + (\sin x - \cos x)^2 = 2\n]", "---", "## Why This Simplification Matters", "Simplifying trigonometric expressions like this helps in:", "- Solving integrals and differential equations\n- Proving identities efficiently\n- Improving computational accuracy in physics and engineering", "Understanding this identity provides a quick method to verify related expressions without redoing expand-and-cancel steps each time.", "---", "## Summary", "- Expand both squared terms\n- Add them carefully\n- Cancel the cross terms\n- Apply the identity (\sin^2 x + \cos^2 x = 1)", "[\n\boxed{(\sin x + \cos x)^2 + (\sin x - \cos x)^2 = 2}\n]", "Take away: With a systematic approach and knowledge of core identities, even complex trigonometric expressions reduce cleanly to fundamental truths.", "---", "## Further Reading", "- Explore more trigonometric identities and their derivations\n- Learn how these simplifications apply in calculus and physics\n- Practice applying similar algebra-identity hybrids in complex expressions", "---", "Keywords: simplify ((\sin x + \cos x)^2 + (\sin x - \cos x)^2), trigonometric identities, algebraic simplification, Pythagorean identity, math example, calculus tips, algebra and trigonometry, mathematical simplification."]
