Question

Tan20+tan40+tan60+......+tan180=?

Answer 1

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.