Express sine of 1/2 angle of a spherical triangle as a function of the sides derivation

Question

In reading through Spherical Trigonometry (Todhunter) he makes a jump from

$1 - \dfrac{\cos a - \cos b \cos c}{\sin b \sin c}$ being equal to $\dfrac{\cos(b - c) - \cos a}{\sin b \sin c}$.

I do not understand how he equates the numerators?

Any help would be appreciated.

Answer 1

Compare the numerators:

$ \cos b \cos c + \sin b \sin c == \cos(b - c)$ an identity from plane trigonometry that has nothing to do with half angles.