Problem: If $\tan x+\tan y=25$ and $\cot x + \cot y=30$, what is $\tan(x+y)$? Solution: Notice that $\cot x + \cot y = \frac{\tan x +\tan y}{\tan x \tan y} = 30$ and since $\tan x + \tan y = 25$ this means $\tan x \tan y = \frac{5}{6}$. Finally, $\tan(x+y) = \frac{\tan x + \tan y}{1-\tan x \tan y} = 150$.