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  "body": "\n## Statement\n\nFor every real $x$,\n$$\\cos^2 x = \\frac{1 + \\cos 2x}{2}, \\qquad \\sin^2 x = \\frac{1 - \\cos 2x}{2}.$$\n\n## Remarks\n\nThe [[addition formula|028XEQCE88]] gives $\\cos 2x = \\cos^2 x - \\sin^2 x$. With the [[Pythagorean identity|03H7WZ4T3K]], $\\cos 2x = 2 \\cos^2 x - 1 = 1 - 2 \\sin^2 x$.\nThese formulas give $\\int_0^{\\pi} \\cos^2 x \\, dx = \\frac{\\pi}{2}$.\nMathlib states the first one as `Real.cos_sq`.\n\n## Explanations\n\n### L1 human\n\nA square of a cosine is a cosine of the double angle, shifted and scaled.\nThis turns squares into first powers, which are easy to integrate.\n",
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