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  "body": "\n## Definition\n\nThe tangent function is defined by\n$$\\tan x = \\frac{\\sin x}{\\cos x}$$\nfor all real $x$ with $\\cos x \\ne 0$, that is, for $x \\ne \\frac{\\pi}{2} + k \\pi$ with $k \\in \\mathbb{Z}$.\n\n## Remarks\n\nThe function is odd and periodic with period $\\pi$.\nOn $(-\\pi/2, \\pi/2)$ it is strictly increasing, and it maps this interval onto $\\mathbb{R}$.\nThe related functions are $\\cot x = \\cos x / \\sin x$, $\\sec x = 1 / \\cos x$ and $\\csc x = 1 / \\sin x$. Mathlib defines `Real.tan` with the value $0$ where $\\cos x = 0$.\n\n## Explanations\n\n### L1 human\n\nThe tangent of an angle is the slope of the line through the origin at that angle.\nIt is undefined where this line is vertical.\n",
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