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HMMT February

Algebra Difficulty 4.9 AIME Find the answer

If tan⁡x+tan⁡y=4\tan x+\tan y=4 and cot⁡x+cot⁡y=5\cot x+\cot y=5, compute tan⁡(x+y)\tan (x+y).

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We have cot⁡x+cot⁡y=tan⁡x+tan⁡ytan⁡xtan⁡y\cot x+\cot y=\frac{\tan x+\tan y}{\tan x \tan y}, so tan⁡xtan⁡y=45\tan x \tan y=\frac{4}{5}. Thus, by the tan⁡\tan sum formula, tan⁡(x+y)=tan⁡x+tan⁡y1−tan⁡xtan⁡y=20\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y}=20.

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