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  "body": "\n## Statement\n\nThe tangent function is differentiable at every $x$ with $\\cos x \\ne 0$, and\n$$\\tan'(x) = \\frac{1}{\\cos^2 x} = 1 + \\tan^2 x.$$\n\n## Remarks\n\nThe [[quotient rule|0BS0NJPKVE]] gives $\\tan' = \\frac{\\cos \\cdot \\cos - \\sin \\cdot (-\\sin)}{\\cos^2}$. The [[Pythagorean identity|03H7WZ4T3K]] turns the numerator into $1$.\nThe derivative is positive. So $\\tan$ is strictly increasing on $(-\\pi/2, \\pi/2)$.\nMathlib states it as `Real.hasDerivAt_tan`.\n\n## Explanations\n\n### L1 human\n\nThe tangent function always rises, and it rises faster and faster near its asymptotes.\nIts slope at $0$ is $1$.\n",
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