+118608 Hi Old Timer,
We are no ignoring your question (part 2 that is). Chris and I just have not worked out how to do it yet. 
Hopefully Heueka or Alan or Tiggsy or some other clever member or guest will show us all how it is done (hopefully in a nice easy to follow fashion.)
Edit:
I hope I haven't offended anyone by using names. Maybe Rom or ElectricPavlov or Omi or geno or ... We have lots of very knowledgable people here now days. 
+239 Hi Melody, Thanks for your update....you are very always helpful...I can vouch for that!
I have a partial solution which I will post later on tomorrow (loading is a bit tough)...but need to rework as in desperation I worked back from the answer, but still not perfect.
+118608 So far I have not worked out how to derive those initial formulas that were given.
I have not looked at the rest of your logic yet, I'd really like to convince myself of the accuracy of the book extract first :/
+26367 Trigonometric Identities - 2
\(\text{Formula } 1: \boxed{\sin(A+B) = \dfrac{2\cdot \tan \left(\frac{A+B}{2} \right)} {1+\tan^2\left(\frac{A+B}{2} \right) } }\\ \text{Formula } 2: \boxed{\cos(A)-\cos(B) = -2\sin\left(\frac{A+B}{2} \right)\sin\left(\frac{A-B}{2} \right) } \\ \text{Formula } 3: \boxed{\sin(A)-\sin(B) = 2\cos\left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2} \right) } \)
\(\begin{array}{|rcll|} \hline \dfrac{p}{q} &=& \dfrac{\cos(A)-\cos(B) } {\sin(A)-\sin(B) } \\\\ &=& \dfrac{-2\sin\left(\frac{A+B}{2} \right)\sin\left(\frac{A-B}{2} \right) } { 2\cos\left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2} \right) } \\\\ &=& - \tan \left(\frac{A+B}{2} \right) \\ \mathbf{\tan \left(\frac{A+B}{2} \right)} &\mathbf{=}& \mathbf{- \dfrac{p}{q}} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \sin(A+B) &=& \dfrac{2\cdot \tan \left(\frac{A+B}{2} \right)} {1+\tan^2\left(\frac{A+B}{2} \right) } \\\\ &=& \dfrac{2\cdot \left( \dfrac{-p}{q} \right) } {1+\left( \dfrac{-p}{q} \right)^2 } \\\\ &=& -\dfrac{2\cdot p } {q\left(1+ \dfrac{p^2}{q^2} \right) } \\\\ &=& -\dfrac{2\cdot p } { q+ \dfrac{p^2}{q } } \\\\ \mathbf{\sin(A+B)} &\mathbf{=}& \mathbf{-\dfrac{2pq } { p^2+q^2 } } \\ \hline \end{array}\)
+118608 Thanks very much Heureka,
Those formulae 2 and 3 that you have used, do you have those memorised?
I mean can they be derived easily enough?
I do not remember seeing them before....
+26367 Hi Melody
For Example:
\(\begin{array}{|lrclrcl|} \hline (1) & \sin(x+y) &=& \sin(x)\cos(y) +\cos(x)\sin(y) \\ (2) & \sin(x-y) &=& \sin(x)\cos(y) -\cos(x)\sin(y) \\ \\ \hline \\ (1)-(2): & \sin(x+y)-\sin(x-y) &=& \sin(x)\cos(y) +\cos(x)\sin(y) \\ &&&-\Big(\sin(x)\cos(y) -\cos(x)\sin(y)\Big) \\ & \sin(x+y)-\sin(x-y) &=& \sin(x)\cos(y) +\cos(x)\sin(y) \\ &&& -\sin(x)\cos(y) +\cos(x)\sin(y) \\ & \sin(x+y)-\sin(x-y) &=& \cos(x)\sin(y) +\cos(x)\sin(y) \\ & \sin(x+y)-\sin(x-y) &=& 2\cos(x)\sin(y) \\ \\ \hline \\ (3): \mathbf{x+y = A} \\ (4): \mathbf{x-y = B} \\ \\ (3)+(4): & 2x = A+B \\ & \mathbf{x = \dfrac{A+B}{2}} \\\\ (3)-(4): & 2y = A-B \\ & \mathbf{ y = \dfrac{A-B}{2}} \\\\ \hline & \sin(x+y)-\sin(x-y) &=& 2\cos(x)\sin(y) \\ & \sin(A)-\sin(B) &=& 2\cos\left(\dfrac{A+B}{2}\right)\sin\left(\dfrac{A-B}{2} \right) \\ \hline \end{array}\)
