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  "body": "\n## Statement\n\nFor every real number $x$ and every integer $n$,\n$$(\\cos x + i \\sin x)^n = \\cos(n x) + i \\sin(n x).$$\n\n## Remarks\n\nProof: by [[Euler's formula|04R3CY41VJ]], the left side is $(e^{i x})^n = e^{i n x}$, which is the right side.\nFor $n = 2$, comparing real and imaginary parts gives $\\cos 2x = \\cos^2 x - \\sin^2 x$ and $\\sin 2x = 2 \\sin x \\cos x$.\nFor non-integer exponents $n$, the formula fails in general.\n\n## Explanations\n\n### L1 human\n\nRaising a point of the unit circle to the power $n$ multiplies its angle by $n$.\nFor example, $\\cos(\\pi/4) + i \\sin(\\pi/4)$ points at $45$ degrees, and its fourth power points at $180$ degrees, so it is $-1$.\n",
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