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