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  "body": "\n## Proof\n\nWe expand the right sides with the definitions and compare.\n\n1. Let $x$ and $y$ be real numbers. Write $e^t = \\exp(t)$ for the [[exponential function|0RMVS7RXZX]].\n2. Let $p = e^x e^y$, $q = e^x e^{-y}$, $r = e^{-x} e^y$ and $s = e^{-x} e^{-y}$.\n3. By the [[exponential of a sum|0E8HNVBCSP]], $p = e^{x + y}$ and $s = e^{-(x + y)}$.\n4. By the definitions of the [[hyperbolic sine|0GWD21EWRW]] and the [[hyperbolic cosine|0EQFWKJ37J]], $\\sinh t = \\frac{e^t - e^{-t}}{2}$ and $\\cosh t = \\frac{e^t + e^{-t}}{2}$ for every real $t$.\n5. By step 4, $4 \\sinh x \\cosh y = (e^x - e^{-x})(e^y + e^{-y}) = p + q - r - s$.\n6. By step 4, $4 \\cosh x \\sinh y = (e^x + e^{-x})(e^y - e^{-y}) = p - q + r - s$.\n7. Adding steps 5 and 6 gives $4(\\sinh x \\cosh y + \\cosh x \\sinh y) = 2p - 2s$.\n8. By steps 3 and 4, $2p - 2s = 2(e^{x+y} - e^{-(x+y)}) = 4 \\sinh(x + y)$. With step 7 this proves $\\sinh(x + y) = \\sinh x \\cosh y + \\cosh x \\sinh y$.\n9. By step 4, $4 \\cosh x \\cosh y = (e^x + e^{-x})(e^y + e^{-y}) = p + q + r + s$.\n10. By step 4, $4 \\sinh x \\sinh y = (e^x - e^{-x})(e^y - e^{-y}) = p - q - r + s$.\n11. Adding steps 9 and 10 gives $4(\\cosh x \\cosh y + \\sinh x \\sinh y) = 2p + 2s$.\n12. By steps 3 and 4, $2p + 2s = 2(e^{x+y} + e^{-(x+y)}) = 4 \\cosh(x + y)$. With step 11 this proves $\\cosh(x + y) = \\cosh x \\cosh y + \\sinh x \\sinh y$.\n\n## Idea\n\nEach product of two hyperbolic functions expands into four products of exponentials. In the right combination, the mixed terms $e^{x - y}$ and $e^{y - x}$ cancel. The surviving terms $e^{x+y}$ and $e^{-(x+y)}$ assemble into the hyperbolic function of $x + y$.\n",
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