Emmanuel José García's user avatar
Emmanuel José García's user avatar
Emmanuel José García's user avatar
Emmanuel José García
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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...

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A transformation that simplifies certain trigonometric integrals and involves the tangent of the half-angle:

$$\int f\left(\tan{\left(\frac12\csc^{-1}(x)\right)}, \tan{\left(\frac12\sec^{-1}(x)\right)} \right)\,dx=\int f\left(e^{i\cos^{-1}(x)}, \frac{1-e^{i\cos^{-1}(x)}}{1+e^{i\cos^{-1}(x)}}\right)\,dx.$$

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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