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  "body": "\n## Proof\n\n1. Let $x$ be real. The [[addition formula for cosine|028XEQCE88]] with $y = x$ gives $\\cos 2x = \\cos x \\cos x - \\sin x \\sin x = \\cos^2 x - \\sin^2 x$.\n2. By the [[Pythagorean identity|03H7WZ4T3K]], $\\sin^2 x = 1 - \\cos^2 x$. Substituting in step 1 gives $\\cos 2x = 2 \\cos^2 x - 1$.\n3. Solving for $\\cos^2 x$ gives $\\cos^2 x = \\frac{1 + \\cos 2x}{2}$.\n4. Also by the Pythagorean identity, $\\cos^2 x = 1 - \\sin^2 x$. Substituting in step 1 gives $\\cos 2x = 1 - 2 \\sin^2 x$.\n5. Solving for $\\sin^2 x$ gives $\\sin^2 x = \\frac{1 - \\cos 2x}{2}$.\n\n## Idea\n\nThe double angle formula writes $\\cos 2x$ as $\\cos^2 x - \\sin^2 x$. The Pythagorean identity lets us remove either square, and then we solve for the other one.\n",
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