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  "body": "\n## Definition\n\nThe *sine and cosine addition equations* are the system of [[functional equations|SRAN4EKWS0]]\n$$S(x + y) = S(x) C(y) + C(x) S(y), \\qquad C(x + y) = C(x) C(y) - S(x) S(y) \\quad \\text{for all } x, y \\in \\R,$$\nfor two unknown [[functions|0EPAT51DXQ]] $S, C \\colon \\R \\to \\R$.\nThe first equation is the *sine addition equation* and the second is the *cosine addition equation*.\nA solution is a pair $(S, C)$ that satisfies both identities for all real $x$ and $y$.\n\n## Remarks\n\nExample: $S = \\sin$ and $C = \\cos$ form a solution, by the angle addition formulas of [[sine and cosine|08WFZP6P89]].\nNon-example: $S(x) = x$, $C(x) = 1$ is not a solution, since the cosine equation gives $1 = 1 - xy$, which fails for $x = y = 1$.\nSet $E = C + iS \\colon \\R \\to \\C$.\nThe two equations together say exactly that $E(x + y) = E(x) E(y)$.\nSo the continuous solutions are $S = C = 0$ and $S(x) = e^{bx} \\sin(ax)$, $C(x) = e^{bx} \\cos(ax)$ with $a, b \\in \\R$.\nEach of the two equations is also studied alone, with one or both functions unknown.\n",
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