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According to wolfram alpha, $\dfrac {\sin \theta} {1 - \cos \theta} = \cot \left(\dfrac{\theta}{2} \right)$.

But how would you get to $\cot \left(\dfrac{\theta}{2} \right)$ if you're given $\dfrac {\sin \theta} {1 - \cos \theta}$?

4 Answers 4

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There is a very standard way of transforming trigonometric functions into rational functions of a different variable - generally first learned when integrating functions, but heavily related to the unit circle and illustrative of basic concepts in algebraic geometry.

One sets $t=\tan \frac \theta 2$, when $\sin \theta = \frac {2t} {1+t^2} $ and $\cos \theta = \frac {1-t^2}{1+t^2}$

This gives the result easily, and is worth knowing.

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    For searching purposes: this is the *Weierstrass substitution*.2012-07-10
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Proof without words: $\displaystyle\cot(\theta/2)=\frac{\color{green}{\sin(\theta)}}{\color{red}{1-\cos(\theta)}}$

$\hspace{3.5cm}$enter image description here

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    +1. I remember seeing a similar pic somewhere on this website which was also posted by you. Am I right or am I dreaming?2012-06-20
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    @Marvis: I don't know [what you mean](http://math.stackexchange.com/a/126075) ;-)2012-06-20
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    Ya. Right that was the pic! Good to know that I am not imagining things :).2012-06-20
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    To nitpick, you need a different picture for $\theta \notin [-\pi/2, \pi/2]$2012-06-20
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    @Marvis: actually, I think this [still works](http://i.stack.imgur.com/IHP9p.png).2012-06-20
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    Nice. Wish I could up-vote more than once.2012-06-20
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All you need are the following formulas:

$$\sin(2A) = 2 \sin(A) \cos(A) \text{ and } \cos(2A) = 1 - 2 \sin^2(A)$$ Now take $A = \theta/2$ to get that $$\sin(\theta) = 2 \sin(\theta/2) \cos(\theta/2) \text{ and } \cos(\theta) = 1 - 2 \sin^2(\theta/2)$$ Now plugging this into your equation, we get that $$\dfrac{\sin(\theta)}{1-\cos(\theta)} = \dfrac{2 \sin(\theta/2) \cos(\theta/2)}{2 \sin^2(\theta/2)} = \dfrac{\cos(\theta/2)}{\sin(\theta/2)} = \cot(\theta/2)$$

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    Oh yeah, thanks. Seems obvious now.2012-06-20
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Alternately, use these formulas: $$\cos(\theta)=\frac{e^{i\theta}+e^{-i\theta}}{2} \mbox{and} \, \sin(\theta)=\frac{e^{i\theta}-e^{-i\theta}}{2i} \, .$$

So \begin{eqnarray*} \frac{\sin(\theta)}{1-\cos(\theta)} & = & \frac{(e^{i \theta} - e^{-i \theta})/(2i)}{(2-e^{i \theta}-e^{-i\theta})/2}\\ & = & \frac{1}{i}\frac{(e^{i \theta/2} +e^{-i\theta/2})(e^{i\theta/2}-e^{-i\theta/2})}{-(e^{i \theta/2}-e^{-i\theta/2})^2}\\ & = & i\frac{e^{i \theta/2}+e^{-i\theta/2}}{e^{i\theta/2}-e^{-i\theta/2}}\\ & =& i \frac{2\cos (\theta/2)}{2i \sin (\theta/2)}\\ & =& \cot(\theta/2) \, .\end{eqnarray*}