Solution: First, expand $ (\sin x + \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x = 1 + 2\sin x \cos x $.
["Title: Mastering Trigonometric Identities: Expanding $ (\sin x + \cos x)^2 $ Step-by-Step", "Meta Description:\nLearn how to expand $ (\sin x + \cos x)^2 $ using foundational trigonometric identities—easily boost your math confidence and understand key calculus and calculus prep concepts.", "---", "Understanding how to expand $ (\sin x + \cos x)^2 $ is one of the fundamental building blocks in trigonometry and higher-level mathematics. This identity not only appears in calculus but also forms the basis for simplifying expressions in integrals, derivatives, and wave equations. In this article, we’ll break down the expansion step-by-step—and show why it matters.", "### Why Expand $ (\sin x + \cos x)^2 $?", "Trigonometric expansions help simplify complex expressions, make integration and differentiation easier, and reveal hidden patterns in periodic functions. By learning this identity, you empower yourself to tackle more advanced problems in physics, engineering, and signal processing.", "---", "### Step 1: Start with the Square of a Sum", "We begin by recognizing that $ (\sin x + \cos x)^2 $ means:", "$$\n(\sin x + \cos x)^2 = (\sin x + \cos x)(\sin x + \cos x)\n$$", "This is the standard algebraic expansion of a binomial squared.", "---", "### Step 2: Apply the Distributive Property (FOIL)", "Use the FOIL method (First, Outer, Inner, Last) to multiply:", "- First: $ \sin x \cdot \sin x = \sin^2 x $\n- Outer: $ \sin x \cdot \cos x = \sin x \cos x $\n- Inner: $ \cos x \cdot \sin x = \sin x \cos x $\n- Last: $ \cos x \cdot \cos x = \cos^2 x $", "Add them together:", "$$\n\sin^2 x + \sin x \cos x + \sin x \cos x + \cos^2 x\n$$", "---", "### Step 3: Combine Like Terms", "Notice that $ \sin x \cos x $ appears twice—combine them:", "$$\n\sin^2 x + 2\sin x \cos x + \cos^2 x\n$$", "---", "### Step 4: Apply the Pythagorean Identity", "Recall one of the fundamental trigonometric identities:", "$$\n\sin^2 x + \cos^2 x = 1\n$$", "Use this to replace $ \sin^2 x + \cos^2 x $ with 1:", "$$\n1 + 2\sin x \cos x\n$$", "---", "### ✅ Final Result", "$$\n(\sin x + \cos x)^2 = 1 + 2\sin x \cos x\n$$", "This simplified form is much easier to work with in further calculations—especially in integration, Fourier series, and deriving power-reduced identities.", "---", "### Pro Tips for Mastery", "- Memorize key identities: Besides $ \sin^2 x + \cos^2 x = 1 $, also recall $ \sin(2x) = 2\sin x \cos x $, so $ 2\sin x \cos x = \sin(2x) $.\n- Practice the expansion: Apply this step-by-step technique to similar binomials to build confidence.\n- Use the identity in calculus: This expanded form is vital when computing integrals involving $ \sin x \cos x $ or derivatives of composite trigonometric expressions.", "---", "### Related Concepts You Should Know", "- Double-angle identities\n- Leveraging symmetry in trigonometric functions\n- Integration techniques like substitution and trigonometric identities\n- Derivatives of sine and cosine functions", "---", "Expanding $ (\sin x + \cos x)^2 $ may seem simple, but mastering it unlocks deeper understanding and flag-ship skills for advanced math. Keep practicing—every step strengthens your foundation!", "---", "Keywords for SEO optimization:\n$ (\sin x + \cos x)^2 $ expansion, trigonometric identities, sine and cosine expansion, $ \sin^2 x + \cos^2 x $, $ 2\sin x \cos x = \sin(2x) $, step-by-step trigonometry, math tutorials, calculus preparation, trig identities guide", "Target Audience:\nHigh school and college students, math enthusiasts, calculus learners, PDF study guides, and anyone building a strong foundation in trigonometry."]
