find-arctan-x-arctany-at-form-of-arctan-

Question Number 98187 by abdomathmax last updated on 12/Jun/20

find \arctan (x) + arctany at form of \arctan

Answered by Rio Michael last updated on 12/Jun/20

let \arctan x = u \Rightarrow x = \tan u

and let \tan y = v \Rightarrow y = \tan v

suppose \arctan x + \arctan y = θ

then u + v = θ

\Rightarrow \tan θ = \frac{\tan u + \tan v}{1 - \tan u \tan v}

\Rightarrow \arctan x + \arctan y = \arctan (\frac{x + y}{1 - x y})

Answered by 1549442205 last updated on 12/Jun/20

Putting α = \arctan (x), β = \arctan (y) we get

\tan α = x, \tan β = y \Rightarrow \tan (α + β) = \frac{\tan α + \tan β}{1 - \tan α \tan β}

= \frac{x + y}{1 - xy} \Rightarrow α + β = \arctan (\frac{x + y}{1 - xy})

Answered by mathmax by abdo last updated on 12/Jun/20

we have \tan (arctanx + arctany) = \frac{x + y}{1 - xy} \Rightarrow

arctanx + arctany = \arctan (\frac{x + y}{1 - xy}) if xy \neq 1

if xy = 1 \Rightarrow y = \frac{1}{x} \Rightarrow arctanx + arctany = \bar{+} \frac{π}{2}