tanA+tanB=1+tanAtanB→
(tanA+tanB)/(1-tanAtanB)=(1+tanAtanB)/(1-tanAtanB)=(cosAcosB+sinAsinB)/(cosAcosB-sinAsinB)
=cos(A-B) /cos(A+B)
即tan(A+B)=cos(A-B) /cos(A+B)
→cos(A-B)=sin(A+B)=cos(90°-A-B)
→A-B=90°-A-B
→A=45°.
由条件中A,B的对称性可知,B也等于45°.
则cos(A+B)=cos(90°)=0.