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

1986 AIME Problem 3

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\tan x+\tan y=25

and

\cot x+\cot y=30,

what is

\tan(x+y)?

Answer

Express the sum of the cotangents using

\tan x

and

\tan y

Use the tangent addition formula after finding

\tan x\tan y

Since

\cot x+\cot y =\frac{\tan x+\tan y}{\tan x\tan y},

the given equations yield

\tan x\tan y=\frac{25}{30}=\frac{5}{6}.

Therefore

\begin{aligned} \tan(x+y) &=\frac{\tan x+\tan y} {1-\tan x\tan y}\\ &=\frac{25}{1-\frac56}\\ &=150. \end{aligned}