First, notice that [tex]\tan180^\circ=0[/tex], so the sum is the same as
[tex]\tan20^\circ+\tan40^\circ+\cdots+\tan140^\circ+\tan160^\circ[/tex]
Next, recall the fact that [tex]\tan(180^\circ-x)=-\tan x[/tex], and notice that
[tex]\tan20^\circ=\tan(180^\circ-160^\circ)=-\tan160^\circ[/tex]
[tex]\tan40^\circ=\tan(180^\circ-140^\circ)=-\tan140^\circ[/tex]
and so on, which means the remaining terms cancel and the value of the sum is 0.
