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The half-angle formulas are central!
Context: The theoretical importance of half-angle formulas
And as a picture is worth a thousand words...
A transformation that simplifies certain trigonometric integrals and involves the tangent of the half-angle:
Building a system through which you can account for a variety of facts within a particular field of knowledge is arguably the greatest aspiration of a scholar. What do Spinoza, Darwin, Freud, and Einstein have in common? They were all great systematizers!
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