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Question Number 50397 by Abdo msup. last updated on 16/Dec/18

$$calculateartan\left(2\right)+arctan\left(5\right)+arctan\left(8\right)$$

Answered by mr W last updated on 17/Dec/18

$$x={\tan }^{-1}2+{\tan }^{-1}8$$$$\tan x=\frac{2+8}{1-2\times 8}=-\frac{2}{3}$$$$x=-{\tan }^{-1}\frac{2}{3}$$$$y={\tan }^{-1}5+x={\tan }^{-1}5-{\tan }^{-1}\frac{2}{3}$$$$\tan y=\frac{5-\frac{2}{3}}{1+5\times \frac{2}{3}}=\frac{13}{13}=1$$$$\Rightarrow y=\frac{\pi }{4}$$$$\Rightarrow {\tan }^{-1}2+{\tan }^{-1}5+{\tan }^{-1}8=\frac{\pi }{4}$$

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