Wikipedia's "Tangent half-angle formula" article lists these: \begin{align} \tan\frac\theta2 & = \frac{\sin\theta}{1+\cos\theta} = \frac{1-\cos\theta}{\sin\theta} = \frac{\tan\theta}{1 + \sec\theta} = \frac{1-\sec\theta}{\tan\theta} \\[10pt] \tan\left(\frac{\eta+\theta}{2}\right) & = \frac{\sin\eta+\sin\theta}{\cos\eta+\cos\theta} \\[10pt] \tan\left(\frac\pi4 \pm \frac\theta2\right) & = \sec\theta\pm\tan\theta \\[10pt] \sqrt{\frac{1-\sin\theta}{1+\sin\theta}} & = \frac{1-\tan\frac\theta2}{1+\tan\frac\theta2} \end{align}

I found it convenient in doing a geometry problem to use this item, which I derived from scratch (it's routine): $$ e^{i\theta} = \frac{1+i\tan\frac\theta2}{1-i\tan\frac\theta2}.\tag{1} $$

So my questions are:

  • Is $(1)$ in "the literature"? (I'm guessing it's in Euler somewhere.)
  • Is this in recent textbooks or the like (say published after 1850)?
  • Are there known to be certain contexts in which its use is convenient?

(If there's a book or a paper that it's in, I might just add it to the Wikipedia article and cite that.)


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