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?