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.