Trig. Identities......

Once again,  that is an impressive effort  Heureka   

Can anybody derive this expression:

arctan(a1/b1) + arctan(a2/b2) = arctan[a1b2 + a2b1] / [b1b2 - a1a2],

IF

-Pi/2 < arctan(a1/b1) + arctan(a2/b2) < Pi/2 {nl} From these two? 

Sin(a + b) = SinaCosb + CosaSinb, and

Cos(a + b) = CosaCosb - SinaSinb

\(\begin{array}{|lrcll|} \hline \text{Let} & a &=& \arctan(\frac{a_1}{b_1}) \\ \text{Let} & b &=& \arctan(\frac{a_2}{b_2}) \\ \hline \end{array}\)

\(\small{ \begin{array}{rcll} \sin(a + b) &=& \sin(a) \cdot \cos(b) + \cos(a) \cdot \sin(b) \quad |\quad a = \arctan(\frac{a_1}{b_1}) \quad b = \arctan(\frac{a_2}{b_2})\\\\ \sin(\arctan(\frac{a_1}{b_1}) + \arctan(\frac{a_2}{b_2})) &=& \sin(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2})) + \cos(\arctan(\frac{a_1}{b_1})) \cdot \sin(\arctan(\frac{a_2}{b_2})) \\\\ \hline \\ \cos(a + b) &=& \cos(a) \cdot \cos(b) - \sin(a) \cdot \sin(b) \quad |\quad a = \arctan(\frac{a_1}{b_1}) \quad b = \arctan(\frac{a_2}{b_2})\\\\ \cos(\arctan(\frac{a_1}{b_1}) + \arctan(\frac{a_2}{b_2})) &=& \cos(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2})) - \sin(\arctan(\frac{a_1}{b_1})) \cdot \sin(\arctan(\frac{a_2}{b_2})) \\\\ \end{array} }\)

\(\begin{array}{|rcll|} \hline \tan(a+b) &=& \frac{\sin(a+b)}{\cos(a+b)} \quad |\quad a = \arctan(\frac{a_1}{b_1}) \quad b = \arctan(\frac{a_2}{b_2})\\\\ \tan(\arctan(\frac{a_1}{b_1})+\arctan(\frac{a_2}{b_2})) &=& \frac{\sin(\arctan(\frac{a_1}{b_1})+\arctan(\frac{a_2}{b_2}))}{\cos(\arctan(\frac{a_1}{b_1})+\arctan(\frac{a_2}{b_2}))} \\ \tan(\arctan(\frac{a_1}{b_1})+\arctan(\frac{a_2}{b_2})) &=& \frac { \sin(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2})) + \cos(\arctan(\frac{a_1}{b_1})) \cdot \sin(\arctan(\frac{a_2}{b_2})) } { \cos(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2})) - \sin(\arctan(\frac{a_1}{b_1})) \cdot \sin(\arctan(\frac{a_2}{b_2})) } \\ \tan(\arctan(\frac{a_1}{b_1})+\arctan(\frac{a_2}{b_2})) &=& \frac { \frac{ \sin(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2})) + \cos(\arctan(\frac{a_1}{b_1})) \cdot \sin(\arctan(\frac{a_2}{b_2})) } {\cos(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2}))} } { \frac{ \cos(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2})) - \sin(\arctan(\frac{a_1}{b_1})) \cdot \sin(\arctan(\frac{a_2}{b_2})) } {\cos(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2}))} }\\ \tan(\arctan(\frac{a_1}{b_1})+\arctan(\frac{a_2}{b_2})) &=& \frac { \frac{ \sin(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2}))}{\cos(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2}))} + \frac{\cos(\arctan(\frac{a_1}{b_1})) \cdot \sin(\arctan(\frac{a_2}{b_2})) } {\cos(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2}))} } { \frac{ \cos(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2}))}{\cos(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2}))} - \frac{\sin(\arctan(\frac{a_1}{b_1})) \cdot \sin(\arctan(\frac{a_2}{b_2})) } {\cos(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2}))} }\\ \tan(\arctan(\frac{a_1}{b_1})+\arctan(\frac{a_2}{b_2})) &=& \frac { \frac{ \sin(\arctan(\frac{a_1}{b_1}))} {\cos(\arctan(\frac{a_1}{b_1})) } + \frac{\sin(\arctan(\frac{a_2}{b_2})) } {\cos(\arctan(\frac{a_2}{b_2}))} } { 1 - \frac{\sin(\arctan(\frac{a_1}{b_1})) \cdot \sin(\arctan(\frac{a_2}{b_2})) } {\cos(\arctan(\frac{a_1}{b_1})) \cdot \cos(\arctan(\frac{a_2}{b_2}))} }\\ \tan(\arctan(\frac{a_1}{b_1})+\arctan(\frac{a_2}{b_2})) &=& \frac { \tan(\arctan(\frac{a_1}{b_1})) + \tan(\arctan(\frac{a_2}{b_2})) } { 1 - \tan(\arctan(\frac{a_1}{b_1})) \cdot \tan(\arctan(\frac{a_2}{b_2})) }\\ \tan(\arctan(\frac{a_1}{b_1})+\arctan(\frac{a_2}{b_2})) &=& \frac { \frac{a_1}{b_1} + \frac{a_2}{b_2} } { 1 - \frac{a_1}{b_1} \cdot \frac{a_2}{b_2} }\\ \tan(\arctan(\frac{a_1}{b_1})+\arctan(\frac{a_2}{b_2})) &=& \frac { \frac{ a_1b_2+a_2b_1}{b_1b_2} } { \frac{ b_1b_2-a_1a_2}{b_1b_2} }\\ \tan(\arctan(\frac{a_1}{b_1})+\arctan(\frac{a_2}{b_2})) &=& \frac{ a_1b_2+a_2b_1 }{ b_1b_2-a_1a_2 } \quad | \quad \arctan() \text{ both sides }\\\\ \mathbf{ \arctan(\frac{a_1}{b_1})+\arctan(\frac{a_2}{b_2}) } & \mathbf{=} & \mathbf{ \arctan(\frac{ a_1b_2+a_2b_1 }{ b_1b_2-a_1a_2 }) } \\ \hline \end{array} \)