Compute-x-if-arctan-x-arctan-1-2-arctan-x-arctan-1-3-

Question Number 127402 by bemath last updated on 29/Dec/20

C o m p u t e x i f \arctan x + \arctan 1 = 2 (\arctan x - \arctan \frac{1}{3})

Answered by liberty last updated on 29/Dec/20

\Leftrightarrow \arctan x + \arctan 1 = 2 \arctan x - 2 \arctan \frac{1}{3}

\Leftrightarrow \arctan x = \arctan 1 + 2 \arctan \frac{1}{3}

\tan (\arctan x) = \tan (\frac{π}{4} + 2 \arctan \frac{1}{3})

x = \frac{1 + \tan (2 \arctan \frac{1}{3})}{1 - \tan (2 \arctan \frac{1}{3})}

(*) \tan (2 \arctan \frac{1}{3}) = \frac{2 \tan (\arctan \frac{1}{3})}{1 - \tan^{2} (\arctan \frac{1}{3})}

= \frac{2 / 3}{1 - 1 / 9} = \frac{3}{4}

\Leftrightarrow x = \frac{1 + \frac{3}{4}}{1 - \frac{3}{4}} = 7 .

Commented by bemath last updated on 29/Dec/20

Danke Meister. Ihre Antwort ist wunderschön