Is it possible to see the steps of TrigFactor, Simplify, and others?

Questions
,
1

I have some very large Trig expressions that I want to analyze. On one of these TrigFactor listed

-2 Sin[\[Beta]/2 - \[Gamma]/2] Sin[\[Beta]/2 + \[Gamma]/2]

then I did

In[56]:= Simplify[-2 Sin[\[Beta]/2 - \[Gamma]/2] Sin[\[Beta]/
    2 + \[Gamma]/2]]

Out[56]= Cos[\[Beta]] - Cos[\[Gamma]]

Great. The final term is much more useful. I’m puzzled how Mathematica derived the first version of the factor. Did it find the latter result and then convert it to the first? There is nothing in the original expression that would call for half-angles. I would like to see derivation chain of this and other Trig manipulations.

2

Short: Mathematica can’t

However: Wolfram Alpha might

Alternatively the following gives some insight. When you have an expression that is just the sum of sines and cosines and powers, generally converting to exponentials and (re)combining them gives simpler expressions:

-2 Sin[\[Beta]/2 - \[Gamma]/2] Sin[\[Beta]/2 + \[Gamma]/2]
TrigToExp[%]
ExpToTrig[%]