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  "body": "\n## Statement\n\nFor every real $x$,\n$$\\cosh^2 x - \\sinh^2 x = 1.$$\n\n## Remarks\n\nThe left side factors as $(\\cosh x - \\sinh x)(\\cosh x + \\sinh x) = e^{-x} e^{x} = 1$.\nSo the point $(\\cosh t, \\sinh t)$ lies on the right branch of the hyperbola $x^2 - y^2 = 1$.\nDividing by $\\cosh^2 x$ gives $1 - \\tanh^2 x = \\operatorname{sech}^2 x$. Mathlib states the identity as `Real.cosh_sq_sub_sinh_sq`.\n\n## Explanations\n\n### L1 human\n\nThis is the hyperbolic version of $\\cos^2 x + \\sin^2 x = 1$, with a minus sign.\nIt follows directly from the definitions with $e^x$ and $e^{-x}$.\n",
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