Tangents & Cotangents

Let $X = \tan x$ and let $Y = \tan y$ .

We are given that:

$X + Y = 4$ and $\dfrac {1}{X} + \dfrac {1}{Y} = 5$ .

We want $\tan (x + y) = \dfrac {\tan x + \tan y}{1 - \tan x \tan y} = \dfrac {X+Y}{1 - XY}$ .

From the second equation above, we have:

$\dfrac {X+Y}{XY} = 5 \rightarrow \dfrac {4}{XY} = 5 \rightarrow XY = \dfrac {4}{5}$ .

So, our answer is:

$\tan (x+y) = \dfrac {X+Y}{1-XY} = \dfrac {4}{1 - \dfrac {4}{5}} = \dfrac {4}{\dfrac {1}{5}} = 20$