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  "body": "\n## Summary\n\nThe tangent half-angle substitution sets $t = \\tan(x/2)$ in an integral. It turns each rational expression in $\\sin x$ and $\\cos x$ into a rational function of $t$. Partial fractions then finish the computation.\n\n## Remarks\n\nThe substitution gives $\\sin x = \\frac{2t}{1 + t^2}$, $\\cos x = \\frac{1 - t^2}{1 + t^2}$ and $dx = \\frac{2 \\, dt}{1 + t^2}$.\nIt works on intervals where $x/2$ stays inside $(-\\pi/2, \\pi/2)$.\nFor example, it gives $\\int \\frac{dx}{\\sin x} = \\ln |\\tan(x/2)| + C$.\n\n## Explanations\n\n### L1 human\n\nThis substitution turns trigonometric integrals into integrals of fractions of polynomials.\nThose integrals are always solvable by partial fractions.\n",
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