How $v=(-\sin\frac{\alpha}{2}\sin\frac{\beta}{2},\cos\frac{\alpha}{2}\sin\frac{\beta}{2},\sin\frac{\alpha}{2}\cos\frac{\beta}{2})$ is derived from... In the book Quarternion and Rotation Sequences, I can't seem to work out how the final equation (colored in $\color{red}{red}$) is derived from the original equation (colored in $\color{blue}{blue}$).
I did check using Wikipedia's list of trigonometric identities as my references as well as the book errata but to no avail.
I copy/paste the text that is giving me problem literally below:

Thus our tracking transformation has axis of rotation given by
$$\color{blue}{v=\left(k,\frac{k\sin\alpha}{\cos\alpha-1},\frac{k\sin\beta}{\cos\beta-1}\right).}$$
Notice that in this computation we determine only the direction of the axis of rotation.
Should we wish to obtain a specific vector as the axis of rotation we may,
for instance, choose $k = -1$, to obtain
$$v=\left(-1,\frac{\sin\alpha}{1-\cos\alpha},\frac{\sin\beta}{1-cos\beta}\right).$$
We note that by using the trigonometric identity
$$1-\cos\alpha=2\sin^{2}\frac{\alpha}{2}$$
we may write the following expression for the axis of the rotation
$$\color{red}{v=\left(-\sin\frac{\alpha}{2}\sin\frac{\beta}{2},\cos\frac{\alpha}{2}\sin\frac{\beta}{2},\sin\frac{\alpha}{2}\cos\frac{\beta}{2}\right)}$$  
Anyone have any idea?  
 A: Facts:


*

*$1-\cos{2\theta}=2\sin^2\frac{\theta}{2}$.

*$\sin{\theta}=2\sin\frac{\theta}{2}\cos\frac{\theta}{2}$, which we can rearrange to $\displaystyle\frac{\sin{\theta}}{2\sin\frac{\theta}{2}}=\cos\frac{\theta}{2}$.


Steps:
So, starting with $$\color{blue}{\vec{v}=\left(k,\frac{k\sin\alpha}{\cos\alpha-1},\frac{k\sin\beta}{\cos\beta-1}\right)}.$$
Since it doesn't matter what $k$ we use, pick $k=-1$ and this becomes
$$\vec{v}=\left(-1,\frac{\sin{\alpha}}{1-\cos{\alpha}},\frac{\sin{\beta}}{1-\cos{\beta}}\right).$$
We can use fact 1 to replace the denominators of the second and third components, obtaining
$$\vec{v}=\left(-1,\frac{\sin{\alpha}}{2\sin^2\frac{\alpha}{2}},\frac{\sin{\beta}}{2\sin^2\frac{\beta}{2}}\right).$$
Now multiply through by $\sin\frac{\alpha}{2}\sin\frac{\beta}{2}$ and we get
$$\vec{v}=\left(-\sin\frac{\alpha}{2}\sin\frac{\beta}{2},\frac{\sin\alpha\sin\frac{\beta}{2}}{2\sin\frac{\alpha}{2}},\frac{\sin{\beta}\sin\frac{\alpha}{2}}{2\sin\frac{\beta}{2}}\right).$$
Finally, we use fact 2 on the second and third terms.
$$\color{red}{\vec{v}=\left(-\sin\frac{\alpha}{2}\sin\frac{\beta}{2},\cos\frac{\alpha}{2}\sin\frac{\beta}{2},\cos\frac{\beta}{2}\sin\frac{\alpha}{2}\right)}.$$
