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 composite trigonometric integrals and serves as an alternative method to trigonometric substitution integration:

$$\int f\left(x,\tan{\frac{\beta}{2}}, \tan{\frac{\gamma}{2}} \right)\,dx=\int f\left(\frac{e^{i\alpha}+e^{-i\alpha}}{2}a, e^{i\alpha}, \frac{1-e^{i\alpha}}{1+e^{i\alpha}}\right)\,\frac{e^{-i\alpha}-e^{i\alpha}}{2i}a\,d\alpha$$

Where $\alpha=\cos^{-1}\left(\frac{x}{a}\right)$, $\beta=\csc^{-1}\left(\frac{x}{a}\right)$ and $\gamma=\sec^{-1}\left(\frac{x}{a}\right).$

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