{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "01A8JGR4JM"
    },
    {
      "field": "uses",
      "target": "0P4NQD746K"
    },
    {
      "field": "uses",
      "target": "0S049HG1W5"
    }
  ],
  "body": "\n## Statement\n\nFor all real numbers $x$ and $y$,\n$$\\sin(x + y) = \\sin x \\cos y + \\cos x \\sin y,$$\n$$\\cos(x + y) = \\cos x \\cos y - \\sin x \\sin y.$$\n\n## Remarks\n\nOne proof multiplies the power series with the [[Cauchy product|0AN28XQ45Z]] and uses the binomial theorem.\nAnother proof fixes $y$ and shows that the difference of the two sides solves $f'' = -f$ with $f(0) = f'(0) = 0$.\nWith $y = x$, they give $\\sin 2x = 2 \\sin x \\cos x$ and $\\cos 2x = \\cos^2 x - \\sin^2 x$. Mathlib states them as `Real.sin_add` and `Real.cos_add`.\n\n## Explanations\n\n### L1 human\n\nThese formulas compute the sine and cosine of a sum of two angles from the sines and cosines of the angles.\nThey are the most important trigonometric identities.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "33B10"
    ],
    "name": "Angle addition formulas",
    "role": "theorem",
    "sources": [
      {
        "locator": "Angle sum and difference identities",
        "url": "https://en.wikipedia.org/wiki/List_of_trigonometric_identities"
      }
    ],
    "tag": "028XEQCE88",
    "uses": [
      "0WB57D9488",
      "0AN28XQ45Z"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/028XEQCE88/",
  "links": [
    {
      "field": "uses",
      "target": "0WB57D9488"
    },
    {
      "field": "uses",
      "target": "0AN28XQ45Z"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/028XEQCE88.md",
  "msc": [
    "33B10"
  ],
  "name": "Angle addition formulas",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/028/028XEQCE88.md",
  "status": null,
  "tag": "028XEQCE88",
  "uri": "m2://028XEQCE88",
  "url": "https://math2.tamnd.com/wiki/angle-addition-formulas/",
  "writing": {
    "tag": "028XEQCE88",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}