Co $tanA+tanB=\frac{sin(A+B)}{cosA.cosB}=2cot\frac{C}{2}=2tan\frac{A+B}{2}$
$<=>\frac{2sin\frac{A+B}{2}.cos\frac{A+B}{2}}{cosA.cosB}=2\frac{sin\frac{A+B}{2}}{cos\frac{A+B}{2}}$
$<=>2cosA.cosB=2cos^{2}\frac{A+B}{2}$
$<=>2cosA.cosB=cos(A+B)+1$
$<=>2cosA.cosB=cosA.cosB-sinAsinB+1$
$<=>cos(A-B)=1$
$<=>\widehat{A}=\widehat{B}=>dpcm$