Answer
See the step-by-step proof below.
Work Step by Step
Recall $ sin(A+B)=sinAcosB+sinBcosA $ and $ cos(A+B)=cosAcosB-sinAsinB $
We have:
$ tan(A+B)=\frac{sin(A+B)}{cos(A+B)}=\frac{sinAcosB+sinBcosA}{cosAcosB-sinAsinB}$
Divide both the numerator and the denominator by $ cosAcosB $:
$ tan(A+B)=\frac{tanA+tanB}{1-tanAtanB}$ which, is the formula.
