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  "body": "\n## Statement\n\nLet $T_0 = 1$, $T_1 = x$ and $T_{n+1} = 2x \\, T_n - T_{n-1}$ for $n \\ge 1$ be the [[Chebyshev polynomials|M6EAR1JHPP]] of the first kind.\nThen for every integer $n \\ge 0$ and every real $\\theta$,\n$$T_n(\\cos \\theta) = \\cos(n \\theta).$$\nEquivalently, $T_n(x) = \\cos(n \\arccos x)$ for all $x \\in [-1, 1]$.\nMoreover, $T_n$ is the only polynomial with this property.\n\n## Remarks\n\nProof idea: the cases $n = 0$ and $n = 1$ are clear.\nThe sum formula for the [[cosine|CFWMB0S5NK]] gives $\\cos((n + 1)\\theta) + \\cos((n - 1)\\theta) = 2 \\cos \\theta \\cos(n \\theta)$.\nThis is the recursion of the $T_n$ evaluated at $x = \\cos \\theta$, so induction on $n$ gives the identity.\nUniqueness holds because two polynomials that agree on the infinite set $[-1, 1]$ are equal.\n\nExample: $\\cos(3\\theta) = 4 \\cos^3 \\theta - 3 \\cos \\theta$, which matches $T_3 = 4x^3 - 3x$.\n\nMany texts take the identity as the definition of $T_n$ and derive the recursion from it.\nOutside $[-1, 1]$ the same polynomial satisfies $T_n(\\cosh t) = \\cosh(n t)$ for real $t$, by the same induction.\n\n## Explanations\n\n### L1 human\n\nThe cosine of $n$ times an angle can always be written as a polynomial in the cosine of the angle.\nThe Chebyshev polynomial $T_n$ is exactly that polynomial.\nSo on $[-1, 1]$, $T_n$ is a cosine wave seen through the change of variable $x = \\cos \\theta$.\n\n### L2 human\n\nThe identity explains most properties of $T_n$ at once.\nIt gives $\\abs{T_n} \\le 1$ on $[-1, 1]$, the zeros $\\cos \\frac{(2k - 1)\\pi}{2n}$ and the extreme points $\\cos \\frac{k\\pi}{n}$.\nIt also turns expansions in $T_n$ on $[-1, 1]$ into cosine series in $\\theta$, so Fourier methods apply to Chebyshev series.\nThe composition rule $T_m(T_n(x)) = T_{mn}(x)$ follows from $\\cos(m \\cdot n\\theta) = \\cos(mn\\theta)$.\n",
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