证明：
∵tan(a+π/4)=(tana+tanπ/4)/(1-tana*tanπ/4)=(tana+1)/(1-tana)
tan(a+3π/4)=(tana+tan3π/4)/(1-tana*tan3π/4)=(tana-1)/(1+tana)
∴tan(a+π/4)+tan(a+3π/4)
=(tana+1)/(1-tana)+(tana-1)/(1+tana)
=[(tana+1)²-(tana+1)²]/(1-tan²a)
=4tana/(1-tan²a)
∵tan2a=2tana/(1-tan²a)
∴tan(a+π/4)+tan(a+3π/4)=2tan2a