$tg\left(\alpha +\beta \right)=\frac{\sin \left(\alpha +\beta \right)}{\cos \left(\alpha +\beta \right)}=\frac{\sin \alpha \cos \beta +\cos \alpha \sin \beta }{\cos \alpha \cos \beta -\sin \alpha \sin \beta }=\frac{\cos \alpha \left(\frac{\sin \alpha }{\cos \alpha }\right)\cdot \cos \beta +\cos \alpha \cdot \cos \beta \frac{\sin \beta }{\cos \beta }}{\left(\cos \alpha \cos \beta \right)\left(1-\frac{\sin \alpha }{\cos \alpha }\frac{\sin \beta }{\cos \beta }\right)}=\frac{a\cos \alpha \cos \beta +\cos \alpha \cdot \cos \beta \cdot b}{\left(\cos \alpha \cos \beta \right)\left(1-ab\right)}=$  

$=\frac{\left(\cos \alpha \cos \beta \right)\left(a+b\right)}{\left(\cos \alpha \cos \beta \right)\left(1-ab\right)}=\frac{a+b}{1-ab}$