Introductio in analysin infinitorum

FACTORUM TRINOMIALIUM. 127 CAP. IX. $=(1+\frac{2z}{\varpi-g})(1-\frac{2z}{\varpi+g})(1+\frac{2z}{3\varpi-g})(1-\frac{2z}{3\varpi+g})$ $(1+\frac{2z}{5\varpi-g})(1-\frac{2z}{5\varpi+g}) \&c.$ Simili modo altera expressio, si Numerator & Denominator per $1-e^{-c}$ multiplicetur, abit in $\frac{e^x+e^{-x}-e^c-e^{-c}}{2-e^c-e^{-c}}$; quae, facto $c=g\sqrt{-1}$ & $x=z\sqrt{-1}$, dat $\frac{\cos. z-\cos. g}{1-\cos. g}=\frac{\cos. z-\cos.(g-z)}{1-\cos. g}=$ $\cos. z-\frac{\sin. g \cdot \sin. z}{1-\cos. g}=\cos. z-\frac{\sin. z}{\tan. \frac{1}{2}g}$. Erit ergo $\cos. z-\cot. \frac{1}{2}g \cdot \sin. z=$ $1-\frac{z^2}{1.2}-\frac{z^3}{1.2.3}\cot. \frac{1}{ [TRUNCATED]