[tex]\it cosa+cosb=0 \Rightarrow cosb=-cosa\ \ \ \ \ (1)\\ \\ a,\ b\in(0,\ \pi)\backslash\{\dfrac{\pi}{2}\}\ \ \ \ \ (2)\\ \\ (1),\ (2) \Rightarrow b=\pi-a \Rightarrow sinb=sina\ \ \ \ \ (3)\\ \\ \\ tgb=\dfrac{sinb}{cosb}\ \stackrel{(1),(3)}{\Longrightarrow}\ \ tgb=\dfrac{sina}{-cosa} =-\dfrac{sina}{cosa}=-tga\\ \\ \\ tga+tgb=tga+(-tga) =tga-tga=0[/tex]
