AIME 1986 P3

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Problem Statement

If $\tan x + \tan y=25$ and $\cot x + \cot y=30$, what is $\tan(x+y)$?

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Solution

$$\tan x + \tan y=25$$ $$\cot x + \cot y=30$$ $$\dfrac{1}{\tan x} + \dfrac{1}{\tan y} = 30$$ $$\dfrac{\tan x + \tan y}{(\tan x)(\tan y)} = 30 \implies (\tan x)(\tan y) = \dfrac{25}{30}$$ $$\tan(x + y) = \dfrac{\tan x + \tan y}{1 - (\tanx)(\tan y)}$$ $$\tan(x + y) = \dfrac{25}{1 - \dfrac{25}{30}}$$ $$\tan(x + y) = \dfrac{25}{\dfrac{25}{30}} = 150$$